:: MATRLIN2 semantic presentation begin theorem Th1: :: MATRLIN2:1 for K being Field for V being VectSp of K for W1, W2, W12 being Subspace of V for U1, U2 being Subspace of W12 st U1 = W1 & U2 = W2 holds ( W1 /\ W2 = U1 /\ U2 & W1 + W2 = U1 + U2 ) proof let K be Field; ::_thesis: for V being VectSp of K for W1, W2, W12 being Subspace of V for U1, U2 being Subspace of W12 st U1 = W1 & U2 = W2 holds ( W1 /\ W2 = U1 /\ U2 & W1 + W2 = U1 + U2 ) let V be VectSp of K; ::_thesis: for W1, W2, W12 being Subspace of V for U1, U2 being Subspace of W12 st U1 = W1 & U2 = W2 holds ( W1 /\ W2 = U1 /\ U2 & W1 + W2 = U1 + U2 ) let W1, W2, W12 be Subspace of V; ::_thesis: for U1, U2 being Subspace of W12 st U1 = W1 & U2 = W2 holds ( W1 /\ W2 = U1 /\ U2 & W1 + W2 = U1 + U2 ) let U1, U2 be Subspace of W12; ::_thesis: ( U1 = W1 & U2 = W2 implies ( W1 /\ W2 = U1 /\ U2 & W1 + W2 = U1 + U2 ) ) assume that A1: U1 = W1 and A2: U2 = W2 ; ::_thesis: ( W1 /\ W2 = U1 /\ U2 & W1 + W2 = U1 + U2 ) reconsider U12 = U1 /\ U2 as Subspace of V by VECTSP_4:26; A3: the carrier of U12 c= the carrier of (W1 /\ W2) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in the carrier of U12 or x in the carrier of (W1 /\ W2) ) assume x in the carrier of U12 ; ::_thesis: x in the carrier of (W1 /\ W2) then x in U1 /\ U2 by STRUCT_0:def_5; then ( x in U1 & x in U2 ) by VECTSP_5:3; then x in W1 /\ W2 by A1, A2, VECTSP_5:3; hence x in the carrier of (W1 /\ W2) by STRUCT_0:def_5; ::_thesis: verum end; the carrier of (W1 /\ W2) c= the carrier of U12 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in the carrier of (W1 /\ W2) or x in the carrier of U12 ) assume x in the carrier of (W1 /\ W2) ; ::_thesis: x in the carrier of U12 then x in W1 /\ W2 by STRUCT_0:def_5; then ( x in W1 & x in W2 ) by VECTSP_5:3; then x in U12 by A1, A2, VECTSP_5:3; hence x in the carrier of U12 by STRUCT_0:def_5; ::_thesis: verum end; then the carrier of (W1 /\ W2) = the carrier of U12 by A3, XBOOLE_0:def_10; hence W1 /\ W2 = U1 /\ U2 by VECTSP_4:29; ::_thesis: W1 + W2 = U1 + U2 reconsider U12 = U1 + U2 as Subspace of V by VECTSP_4:26; A4: the carrier of (W1 + W2) c= the carrier of U12 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in the carrier of (W1 + W2) or x in the carrier of U12 ) assume x in the carrier of (W1 + W2) ; ::_thesis: x in the carrier of U12 then x in W1 + W2 by STRUCT_0:def_5; then consider v1, v2 being Vector of V such that A5: v1 in W1 and A6: v2 in W2 and A7: v1 + v2 = x by VECTSP_5:1; U2 is Subspace of U12 by VECTSP_5:7; then A8: v2 in U12 by A2, A6, VECTSP_4:8; U1 is Subspace of U12 by VECTSP_5:7; then v1 in U12 by A1, A5, VECTSP_4:8; then reconsider w1 = v1, w2 = v2 as Vector of U12 by A8, STRUCT_0:def_5; v1 + v2 = w1 + w2 by VECTSP_4:13; hence x in the carrier of U12 by A7; ::_thesis: verum end; the carrier of U12 c= the carrier of (W1 + W2) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in the carrier of U12 or x in the carrier of (W1 + W2) ) assume x in the carrier of U12 ; ::_thesis: x in the carrier of (W1 + W2) then x in U1 + U2 by STRUCT_0:def_5; then consider v1, v2 being Vector of W12 such that A9: ( v1 in U1 & v2 in U2 & v1 + v2 = x ) by VECTSP_5:1; reconsider w1 = v1, w2 = v2 as Vector of V by VECTSP_4:10; v1 + v2 = w1 + w2 by VECTSP_4:13; then x in W1 + W2 by A1, A2, A9, VECTSP_5:1; hence x in the carrier of (W1 + W2) by STRUCT_0:def_5; ::_thesis: verum end; then the carrier of (W1 + W2) = the carrier of U12 by A4, XBOOLE_0:def_10; hence W1 + W2 = U1 + U2 by VECTSP_4:29; ::_thesis: verum end; theorem Th2: :: MATRLIN2:2 for K being Field for V being VectSp of K for W1, W2 being Subspace of V st W1 /\ W2 = (0). V holds for B1 being linearly-independent Subset of W1 for B2 being linearly-independent Subset of W2 holds B1 \/ B2 is linearly-independent Subset of (W1 + W2) proof let K be Field; ::_thesis: for V being VectSp of K for W1, W2 being Subspace of V st W1 /\ W2 = (0). V holds for B1 being linearly-independent Subset of W1 for B2 being linearly-independent Subset of W2 holds B1 \/ B2 is linearly-independent Subset of (W1 + W2) let V be VectSp of K; ::_thesis: for W1, W2 being Subspace of V st W1 /\ W2 = (0). V holds for B1 being linearly-independent Subset of W1 for B2 being linearly-independent Subset of W2 holds B1 \/ B2 is linearly-independent Subset of (W1 + W2) let W1, W2 be Subspace of V; ::_thesis: ( W1 /\ W2 = (0). V implies for B1 being linearly-independent Subset of W1 for B2 being linearly-independent Subset of W2 holds B1 \/ B2 is linearly-independent Subset of (W1 + W2) ) assume A1: W1 /\ W2 = (0). V ; ::_thesis: for B1 being linearly-independent Subset of W1 for B2 being linearly-independent Subset of W2 holds B1 \/ B2 is linearly-independent Subset of (W1 + W2) reconsider W19 = W1, W29 = W2 as Subspace of W1 + W2 by VECTSP_5:7; let B1 be linearly-independent Subset of W1; ::_thesis: for B2 being linearly-independent Subset of W2 holds B1 \/ B2 is linearly-independent Subset of (W1 + W2) let B2 be linearly-independent Subset of W2; ::_thesis: B1 \/ B2 is linearly-independent Subset of (W1 + W2) A2: W2 is Subspace of W1 + W2 by VECTSP_5:7; then the carrier of W2 c= the carrier of (W1 + W2) by VECTSP_4:def_2; then A3: B2 c= the carrier of (W1 + W2) by XBOOLE_1:1; A4: W1 is Subspace of W1 + W2 by VECTSP_5:7; then the carrier of W1 c= the carrier of (W1 + W2) by VECTSP_4:def_2; then B1 c= the carrier of (W1 + W2) by XBOOLE_1:1; then reconsider B12 = B1 \/ B2, B19 = B1, B29 = B2 as Subset of (W1 + W2) by A3, XBOOLE_1:8; B12 is linearly-independent proof let L be Linear_Combination of B12; :: according to VECTSP_7:def_1 ::_thesis: ( not Sum L = 0. (W1 + W2) or Carrier L = {} ) assume Sum L = 0. (W1 + W2) ; ::_thesis: Carrier L = {} then A5: Sum L = (0. (W1 + W2)) + (0. (W1 + W2)) by RLVECT_1:def_4; set C = (Carrier L) /\ B1; defpred S1[ set ] means $1 in (Carrier L) /\ B1; (Carrier L) /\ B1 c= Carrier L by XBOOLE_1:17; then reconsider C = (Carrier L) /\ B1 as finite Subset of (W1 + W2) by XBOOLE_1:1; set D = (Carrier L) \ B1; deffunc H1( set ) -> set = L . $1; defpred S2[ set ] means $1 in (Carrier L) \ B1; reconsider D = (Carrier L) \ B1 as finite Subset of (W1 + W2) ; A6: D c= B29 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in D or x in B29 ) assume x in D ; ::_thesis: x in B29 then A7: ( x in Carrier L & not x in B19 ) by XBOOLE_0:def_5; Carrier L c= B12 by VECTSP_6:def_4; hence x in B29 by A7, XBOOLE_0:def_3; ::_thesis: verum end; (0). V = (0). (W1 + W2) by VECTSP_4:36; then A8: W19 /\ W29 = (0). (W1 + W2) by A1, Th1; W19 + W29 = W1 + W2 by Th1; then A9: W1 + W2 is_the_direct_sum_of W19,W29 by A8, VECTSP_5:def_4; A10: B29 is linearly-independent by A2, VECTSP_9:11; A11: B19 is linearly-independent by A4, VECTSP_9:11; deffunc H2( set ) -> Element of the carrier of K = 0. K; A12: ( 0. W1 in W19 & 0. W2 in W29 ) by STRUCT_0:def_5; A13: now__::_thesis:_for_x_being_set_st_x_in_the_carrier_of_(W1_+_W2)_holds_ (_(_S1[x]_implies_H1(x)_in_the_carrier_of_K_)_&_(_not_S1[x]_implies_H2(x)_in_the_carrier_of_K_)_) let x be set ; ::_thesis: ( x in the carrier of (W1 + W2) implies ( ( S1[x] implies H1(x) in the carrier of K ) & ( not S1[x] implies H2(x) in the carrier of K ) ) ) assume x in the carrier of (W1 + W2) ; ::_thesis: ( ( S1[x] implies H1(x) in the carrier of K ) & ( not S1[x] implies H2(x) in the carrier of K ) ) then reconsider v = x as Vector of (W1 + W2) ; L . v in the carrier of K ; hence ( S1[x] implies H1(x) in the carrier of K ) ; ::_thesis: ( not S1[x] implies H2(x) in the carrier of K ) assume not S1[x] ; ::_thesis: H2(x) in the carrier of K thus H2(x) in the carrier of K ; ::_thesis: verum end; consider f being Function of the carrier of (W1 + W2), the carrier of K such that A14: for x being set st x in the carrier of (W1 + W2) holds ( ( S1[x] implies f . x = H1(x) ) & ( not S1[x] implies f . x = H2(x) ) ) from FUNCT_2:sch_5(A13); deffunc H3( set ) -> set = L . $1; A15: now__::_thesis:_for_x_being_set_st_x_in_the_carrier_of_(W1_+_W2)_holds_ (_(_S2[x]_implies_H3(x)_in_the_carrier_of_K_)_&_(_not_S2[x]_implies_H2(x)_in_the_carrier_of_K_)_) let x be set ; ::_thesis: ( x in the carrier of (W1 + W2) implies ( ( S2[x] implies H3(x) in the carrier of K ) & ( not S2[x] implies H2(x) in the carrier of K ) ) ) assume x in the carrier of (W1 + W2) ; ::_thesis: ( ( S2[x] implies H3(x) in the carrier of K ) & ( not S2[x] implies H2(x) in the carrier of K ) ) then reconsider v = x as Vector of (W1 + W2) ; L . v in the carrier of K ; hence ( S2[x] implies H3(x) in the carrier of K ) ; ::_thesis: ( not S2[x] implies H2(x) in the carrier of K ) assume not S2[x] ; ::_thesis: H2(x) in the carrier of K thus H2(x) in the carrier of K ; ::_thesis: verum end; consider g being Function of the carrier of (W1 + W2), the carrier of K such that A16: for x being set st x in the carrier of (W1 + W2) holds ( ( S2[x] implies g . x = H3(x) ) & ( not S2[x] implies g . x = H2(x) ) ) from FUNCT_2:sch_5(A15); reconsider g = g as Element of Funcs ( the carrier of (W1 + W2), the carrier of K) by FUNCT_2:8; for u being Vector of (W1 + W2) st not u in D holds g . u = 0. K by A16; then reconsider g = g as Linear_Combination of W1 + W2 by VECTSP_6:def_1; A17: Carrier g c= D proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Carrier g or x in D ) assume x in Carrier g ; ::_thesis: x in D then A18: ex u being Vector of (W1 + W2) st ( x = u & g . u <> 0. K ) ; assume not x in D ; ::_thesis: contradiction hence contradiction by A16, A18; ::_thesis: verum end; then Carrier g c= B29 by A6, XBOOLE_1:1; then reconsider g = g as Linear_Combination of B29 by VECTSP_6:def_4; reconsider f = f as Element of Funcs ( the carrier of (W1 + W2), the carrier of K) by FUNCT_2:8; for u being Vector of (W1 + W2) st not u in C holds f . u = 0. K by A14; then reconsider f = f as Linear_Combination of W1 + W2 by VECTSP_6:def_1; A19: Carrier f c= B19 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Carrier f or x in B19 ) assume x in Carrier f ; ::_thesis: x in B19 then A20: ex u being Vector of (W1 + W2) st ( x = u & f . u <> 0. K ) ; assume not x in B19 ; ::_thesis: contradiction then not x in C by XBOOLE_0:def_4; hence contradiction by A14, A20; ::_thesis: verum end; then reconsider f = f as Linear_Combination of B19 by VECTSP_6:def_4; ex f1 being Linear_Combination of W19 st ( Carrier f1 = Carrier f & Sum f = Sum f1 ) by A19, VECTSP_9:9, XBOOLE_1:1; then A21: Sum f in W19 by STRUCT_0:def_5; A22: L = f + g proof let v be Vector of (W1 + W2); :: according to VECTSP_6:def_7 ::_thesis: L . v = (f + g) . v now__::_thesis:_(f_+_g)_._v_=_L_._v percases ( v in C or not v in C ) ; supposeA23: v in C ; ::_thesis: (f + g) . v = L . v A24: now__::_thesis:_not_v_in_D assume v in D ; ::_thesis: contradiction then not v in B19 by XBOOLE_0:def_5; hence contradiction by A23, XBOOLE_0:def_4; ::_thesis: verum end; thus (f + g) . v = (f . v) + (g . v) by VECTSP_6:22 .= (L . v) + (g . v) by A14, A23 .= (L . v) + (0. K) by A16, A24 .= L . v by RLVECT_1:4 ; ::_thesis: verum end; supposeA25: not v in C ; ::_thesis: L . v = (f + g) . v now__::_thesis:_(f_+_g)_._v_=_L_._v percases ( v in Carrier L or not v in Carrier L ) ; supposeA26: v in Carrier L ; ::_thesis: (f + g) . v = L . v A27: now__::_thesis:_v_in_D assume not v in D ; ::_thesis: contradiction then ( not v in Carrier L or v in B19 ) by XBOOLE_0:def_5; hence contradiction by A25, A26, XBOOLE_0:def_4; ::_thesis: verum end; thus (f + g) . v = (f . v) + (g . v) by VECTSP_6:22 .= (g . v) + (0. K) by A14, A25 .= g . v by RLVECT_1:4 .= L . v by A16, A27 ; ::_thesis: verum end; supposeA28: not v in Carrier L ; ::_thesis: (f + g) . v = L . v then A29: not v in D by XBOOLE_0:def_5; A30: not v in C by A28, XBOOLE_0:def_4; thus (f + g) . v = (f . v) + (g . v) by VECTSP_6:22 .= (0. K) + (g . v) by A14, A30 .= (0. K) + (0. K) by A16, A29 .= 0. K by RLVECT_1:4 .= L . v by A28 ; ::_thesis: verum end; end; end; hence L . v = (f + g) . v ; ::_thesis: verum end; end; end; hence L . v = (f + g) . v ; ::_thesis: verum end; then A31: Sum L = (Sum f) + (Sum g) by VECTSP_6:44; Carrier g c= B2 by A17, A6, XBOOLE_1:1; then ex g1 being Linear_Combination of W29 st ( Carrier g1 = Carrier g & Sum g = Sum g1 ) by VECTSP_9:9, XBOOLE_1:1; then A32: Sum g in W29 by STRUCT_0:def_5; A33: ( 0. (W1 + W2) = 0. W19 & 0. (W1 + W2) = 0. W29 ) by VECTSP_4:11; then Sum f = 0. (W1 + W2) by A31, A21, A32, A9, A12, A5, VECTSP_5:48; then A34: Carrier f = {} by A11, VECTSP_7:def_1; Sum g = 0. (W1 + W2) by A31, A21, A32, A9, A33, A12, A5, VECTSP_5:48; then A35: Carrier g = {} by A10, VECTSP_7:def_1; {} \/ {} = {} ; hence Carrier L = {} by A22, A34, A35, VECTSP_6:23, XBOOLE_1:3; ::_thesis: verum end; hence B1 \/ B2 is linearly-independent Subset of (W1 + W2) ; ::_thesis: verum end; theorem Th3: :: MATRLIN2:3 for K being Field for V being VectSp of K for W1, W2 being Subspace of V st W1 /\ W2 = (0). V holds for B1 being Basis of W1 for B2 being Basis of W2 holds B1 \/ B2 is Basis of W1 + W2 proof let K be Field; ::_thesis: for V being VectSp of K for W1, W2 being Subspace of V st W1 /\ W2 = (0). V holds for B1 being Basis of W1 for B2 being Basis of W2 holds B1 \/ B2 is Basis of W1 + W2 let V be VectSp of K; ::_thesis: for W1, W2 being Subspace of V st W1 /\ W2 = (0). V holds for B1 being Basis of W1 for B2 being Basis of W2 holds B1 \/ B2 is Basis of W1 + W2 let W1, W2 be Subspace of V; ::_thesis: ( W1 /\ W2 = (0). V implies for B1 being Basis of W1 for B2 being Basis of W2 holds B1 \/ B2 is Basis of W1 + W2 ) assume A1: W1 /\ W2 = (0). V ; ::_thesis: for B1 being Basis of W1 for B2 being Basis of W2 holds B1 \/ B2 is Basis of W1 + W2 let B1 be Basis of W1; ::_thesis: for B2 being Basis of W2 holds B1 \/ B2 is Basis of W1 + W2 let B2 be Basis of W2; ::_thesis: B1 \/ B2 is Basis of W1 + W2 A2: W2 is Subspace of W1 + W2 by VECTSP_5:7; then the carrier of W2 c= the carrier of (W1 + W2) by VECTSP_4:def_2; then A3: B2 c= the carrier of (W1 + W2) by XBOOLE_1:1; A4: W1 is Subspace of W1 + W2 by VECTSP_5:7; then the carrier of W1 c= the carrier of (W1 + W2) by VECTSP_4:def_2; then B1 c= the carrier of (W1 + W2) by XBOOLE_1:1; then reconsider B12 = B1 \/ B2, B19 = B1, B29 = B2 as Subset of (W1 + W2) by A3, XBOOLE_1:8; A5: (Omega). W2 = Lin B2 by VECTSP_7:def_3 .= Lin B29 by A2, VECTSP_9:17 ; A6: Lin B12 = (Lin B19) + (Lin B29) by VECTSP_7:15; A7: (Omega). W1 = Lin B1 by VECTSP_7:def_3 .= Lin B19 by A4, VECTSP_9:17 ; A8: the carrier of (W1 + W2) c= the carrier of (Lin B12) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in the carrier of (W1 + W2) or x in the carrier of (Lin B12) ) assume A9: x in the carrier of (W1 + W2) ; ::_thesis: x in the carrier of (Lin B12) reconsider x = x as Vector of (W1 + W2) by A9; x in W1 + W2 by STRUCT_0:def_5; then consider v1, v2 being Vector of V such that A10: v1 in W1 and A11: v2 in W2 and A12: x = v1 + v2 by VECTSP_5:1; ( v1 is Vector of W1 & v2 is Vector of W2 ) by A10, A11, STRUCT_0:def_5; then reconsider w1 = v1, w2 = v2 as Vector of (W1 + W2) by A4, A2, VECTSP_4:10; A13: v1 + v2 = w1 + w2 by VECTSP_4:13; v2 in the carrier of (Lin B29) by A5, A11, STRUCT_0:def_5; then A14: v2 in Lin B29 by STRUCT_0:def_5; v1 in the carrier of (Lin B19) by A7, A10, STRUCT_0:def_5; then v1 in Lin B19 by STRUCT_0:def_5; then w1 + w2 in Lin B12 by A6, A14, VECTSP_5:1; hence x in the carrier of (Lin B12) by A12, A13, STRUCT_0:def_5; ::_thesis: verum end; the carrier of (Lin B12) c= the carrier of (W1 + W2) by VECTSP_4:def_2; then the carrier of (Lin B12) = the carrier of (W1 + W2) by A8, XBOOLE_0:def_10; then A15: Lin B12 = VectSpStr(# the carrier of (W1 + W2), the addF of (W1 + W2), the ZeroF of (W1 + W2), the lmult of (W1 + W2) #) by VECTSP_4:31; ( B2 is linearly-independent & B1 is linearly-independent ) by VECTSP_7:def_3; then B1 \/ B2 is linearly-independent Subset of (W1 + W2) by A1, Th2; hence B1 \/ B2 is Basis of W1 + W2 by A15, VECTSP_7:def_3; ::_thesis: verum end; theorem :: MATRLIN2:4 for K being Field for V being finite-dimensional VectSp of K for B being OrdBasis of (Omega). V holds B is OrdBasis of V proof let K be Field; ::_thesis: for V being finite-dimensional VectSp of K for B being OrdBasis of (Omega). V holds B is OrdBasis of V let V be finite-dimensional VectSp of K; ::_thesis: for B being OrdBasis of (Omega). V holds B is OrdBasis of V let B be OrdBasis of (Omega). V; ::_thesis: B is OrdBasis of V reconsider r = rng B as Basis of (Omega). V by MATRLIN:def_2; r is linearly-independent by VECTSP_7:def_3; then reconsider R = r as linearly-independent Subset of V by VECTSP_9:11; Lin R = Lin r by VECTSP_9:17 .= VectSpStr(# the carrier of V, the addF of V, the ZeroF of V, the lmult of V #) by VECTSP_7:def_3 ; then A1: R is Basis of V by VECTSP_7:def_3; B is one-to-one by MATRLIN:def_2; hence B is OrdBasis of V by A1, MATRLIN:def_2; ::_thesis: verum end; theorem :: MATRLIN2:5 for K being Field for V1 being VectSp of K for A being finite Subset of V1 st dim (Lin A) = card A holds A is linearly-independent proof let K be Field; ::_thesis: for V1 being VectSp of K for A being finite Subset of V1 st dim (Lin A) = card A holds A is linearly-independent let V1 be VectSp of K; ::_thesis: for A being finite Subset of V1 st dim (Lin A) = card A holds A is linearly-independent let A be finite Subset of V1; ::_thesis: ( dim (Lin A) = card A implies A is linearly-independent ) assume A1: dim (Lin A) = card A ; ::_thesis: A is linearly-independent set L = Lin A; A c= the carrier of (Lin A) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in A or x in the carrier of (Lin A) ) assume x in A ; ::_thesis: x in the carrier of (Lin A) then x in Lin A by VECTSP_7:8; hence x in the carrier of (Lin A) by STRUCT_0:def_5; ::_thesis: verum end; then reconsider A9 = A as Subset of (Lin A) ; Lin A9 = Lin A by VECTSP_9:17; then consider B being Subset of (Lin A) such that A2: B c= A9 and A3: B is linearly-independent and A4: Lin B = Lin A by VECTSP_7:18; reconsider B = B as finite Subset of (Lin A) by A2; B is Basis of Lin A by A3, A4, VECTSP_7:def_3; then reconsider L = Lin A as finite-dimensional VectSp of K by MATRLIN:def_1; card A = dim L by A1 .= card B by A3, A4, VECTSP_9:26 ; then A = B by A2, CARD_FIN:1; hence A is linearly-independent by A3, VECTSP_9:11; ::_thesis: verum end; theorem :: MATRLIN2:6 for K being Field for V being VectSp of K for A being finite Subset of V holds dim (Lin A) <= card A proof let K be Field; ::_thesis: for V being VectSp of K for A being finite Subset of V holds dim (Lin A) <= card A let V be VectSp of K; ::_thesis: for A being finite Subset of V holds dim (Lin A) <= card A let A be finite Subset of V; ::_thesis: dim (Lin A) <= card A set L = Lin A; A c= the carrier of (Lin A) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in A or x in the carrier of (Lin A) ) assume x in A ; ::_thesis: x in the carrier of (Lin A) then x in Lin A by VECTSP_7:8; hence x in the carrier of (Lin A) by STRUCT_0:def_5; ::_thesis: verum end; then reconsider A9 = A as Subset of (Lin A) ; Lin A9 = Lin A by VECTSP_9:17; then consider B being Subset of (Lin A) such that A1: B c= A9 and A2: ( B is linearly-independent & Lin B = Lin A ) by VECTSP_7:18; reconsider B = B as finite Subset of (Lin A) by A1; B is Basis of Lin A by A2, VECTSP_7:def_3; then reconsider L = Lin A as finite-dimensional VectSp of K by MATRLIN:def_1; ( card B = dim L & card B c= card A ) by A1, A2, CARD_1:11, VECTSP_9:26; hence dim (Lin A) <= card A by NAT_1:39; ::_thesis: verum end; begin Lm1: for K being Field for V1 being finite-dimensional VectSp of K for R being FinSequence of V1 for p being FinSequence of K holds dom (lmlt (p,R)) = (dom p) /\ (dom R) proof let K be Field; ::_thesis: for V1 being finite-dimensional VectSp of K for R being FinSequence of V1 for p being FinSequence of K holds dom (lmlt (p,R)) = (dom p) /\ (dom R) let V1 be finite-dimensional VectSp of K; ::_thesis: for R being FinSequence of V1 for p being FinSequence of K holds dom (lmlt (p,R)) = (dom p) /\ (dom R) let R be FinSequence of V1; ::_thesis: for p being FinSequence of K holds dom (lmlt (p,R)) = (dom p) /\ (dom R) let p be FinSequence of K; ::_thesis: dom (lmlt (p,R)) = (dom p) /\ (dom R) ( rng p c= the carrier of K & rng R c= the carrier of V1 ) by FINSEQ_1:def_4; then [:(rng p),(rng R):] c= [: the carrier of K, the carrier of V1:] by ZFMISC_1:96; then [:(rng p),(rng R):] c= dom the lmult of V1 by FUNCT_2:def_1; hence dom (lmlt (p,R)) = (dom p) /\ (dom R) by FUNCOP_1:69; ::_thesis: verum end; Lm2: for K being Field for p1, p2 being FinSequence of K holds dom (p1 + p2) = (dom p1) /\ (dom p2) proof let K be Field; ::_thesis: for p1, p2 being FinSequence of K holds dom (p1 + p2) = (dom p1) /\ (dom p2) let p1, p2 be FinSequence of K; ::_thesis: dom (p1 + p2) = (dom p1) /\ (dom p2) ( rng p1 c= the carrier of K & rng p2 c= the carrier of K ) by FINSEQ_1:def_4; then [:(rng p1),(rng p2):] c= [: the carrier of K, the carrier of K:] by ZFMISC_1:96; then [:(rng p1),(rng p2):] c= dom the addF of K by FUNCT_2:def_1; hence dom (p1 + p2) = (dom p1) /\ (dom p2) by FUNCOP_1:69; ::_thesis: verum end; Lm3: for K being Field for V1 being finite-dimensional VectSp of K for R1, R2 being FinSequence of V1 holds dom (R1 + R2) = (dom R1) /\ (dom R2) proof let K be Field; ::_thesis: for V1 being finite-dimensional VectSp of K for R1, R2 being FinSequence of V1 holds dom (R1 + R2) = (dom R1) /\ (dom R2) let V1 be finite-dimensional VectSp of K; ::_thesis: for R1, R2 being FinSequence of V1 holds dom (R1 + R2) = (dom R1) /\ (dom R2) let R1, R2 be FinSequence of V1; ::_thesis: dom (R1 + R2) = (dom R1) /\ (dom R2) ( rng R1 c= the carrier of V1 & rng R2 c= the carrier of V1 ) by FINSEQ_1:def_4; then [:(rng R1),(rng R2):] c= [: the carrier of V1, the carrier of V1:] by ZFMISC_1:96; then [:(rng R1),(rng R2):] c= dom the addF of V1 by FUNCT_2:def_1; hence dom (R1 + R2) = (dom R1) /\ (dom R2) by FUNCOP_1:69; ::_thesis: verum end; theorem Th7: :: MATRLIN2:7 for K being Field for V1 being finite-dimensional VectSp of K for R being FinSequence of V1 for p1, p2 being FinSequence of K holds lmlt ((p1 + p2),R) = (lmlt (p1,R)) + (lmlt (p2,R)) proof let K be Field; ::_thesis: for V1 being finite-dimensional VectSp of K for R being FinSequence of V1 for p1, p2 being FinSequence of K holds lmlt ((p1 + p2),R) = (lmlt (p1,R)) + (lmlt (p2,R)) let V1 be finite-dimensional VectSp of K; ::_thesis: for R being FinSequence of V1 for p1, p2 being FinSequence of K holds lmlt ((p1 + p2),R) = (lmlt (p1,R)) + (lmlt (p2,R)) let R be FinSequence of V1; ::_thesis: for p1, p2 being FinSequence of K holds lmlt ((p1 + p2),R) = (lmlt (p1,R)) + (lmlt (p2,R)) let p1, p2 be FinSequence of K; ::_thesis: lmlt ((p1 + p2),R) = (lmlt (p1,R)) + (lmlt (p2,R)) set L12 = lmlt ((p1 + p2),R); set L1 = lmlt (p1,R); set L2 = lmlt (p2,R); A1: dom ((lmlt (p1,R)) + (lmlt (p2,R))) = (dom (lmlt (p1,R))) /\ (dom (lmlt (p2,R))) by Lm3; A2: dom (lmlt ((p1 + p2),R)) = (dom (p1 + p2)) /\ (dom R) by Lm1; A3: dom (lmlt (p1,R)) = (dom p1) /\ (dom R) by Lm1; A4: dom (lmlt (p2,R)) = (dom p2) /\ (dom R) by Lm1; then A5: dom ((lmlt (p1,R)) + (lmlt (p2,R))) = (((dom p1) /\ (dom R)) /\ (dom p2)) /\ (dom R) by A1, A3, XBOOLE_1:16 .= (((dom p1) /\ (dom p2)) /\ (dom R)) /\ (dom R) by XBOOLE_1:16 .= ((dom p1) /\ (dom p2)) /\ ((dom R) /\ (dom R)) by XBOOLE_1:16 .= dom (lmlt ((p1 + p2),R)) by A2, Lm2 ; now__::_thesis:_for_x_being_set_st_x_in_dom_((lmlt_(p1,R))_+_(lmlt_(p2,R)))_holds_ ((lmlt_(p1,R))_+_(lmlt_(p2,R)))_._x_=_(lmlt_((p1_+_p2),R))_._x let x be set ; ::_thesis: ( x in dom ((lmlt (p1,R)) + (lmlt (p2,R))) implies ((lmlt (p1,R)) + (lmlt (p2,R))) . x = (lmlt ((p1 + p2),R)) . x ) assume A6: x in dom ((lmlt (p1,R)) + (lmlt (p2,R))) ; ::_thesis: ((lmlt (p1,R)) + (lmlt (p2,R))) . x = (lmlt ((p1 + p2),R)) . x A7: x in dom (lmlt (p2,R)) by A1, A6, XBOOLE_0:def_4; then A8: (lmlt (p2,R)) /. x = (lmlt (p2,R)) . x by PARTFUN1:def_6; x in dom p2 by A4, A7, XBOOLE_0:def_4; then A9: p2 /. x = p2 . x by PARTFUN1:def_6; A10: x in dom (p1 + p2) by A2, A5, A6, XBOOLE_0:def_4; then A11: (p1 + p2) . x = (p1 + p2) /. x by PARTFUN1:def_6; A12: x in dom (lmlt (p1,R)) by A1, A6, XBOOLE_0:def_4; then x in dom p1 by A3, XBOOLE_0:def_4; then A13: p1 /. x = p1 . x by PARTFUN1:def_6; x in dom R by A3, A12, XBOOLE_0:def_4; then A14: R /. x = R . x by PARTFUN1:def_6; A15: (lmlt (p1,R)) /. x = (lmlt (p1,R)) . x by A12, PARTFUN1:def_6; hence ((lmlt (p1,R)) + (lmlt (p2,R))) . x = ((lmlt (p1,R)) /. x) + ((lmlt (p2,R)) /. x) by A6, A8, FVSUM_1:17 .= ( the lmult of V1 . ((p1 /. x),(R /. x))) + ((lmlt (p2,R)) /. x) by A12, A15, A13, A14, FUNCOP_1:22 .= ((p1 /. x) * (R /. x)) + ((p2 /. x) * (R /. x)) by A7, A8, A9, A14, FUNCOP_1:22 .= ((p1 /. x) + (p2 /. x)) * (R /. x) by VECTSP_1:def_15 .= ((p1 + p2) /. x) * (R /. x) by A10, A13, A9, A11, FVSUM_1:17 .= (lmlt ((p1 + p2),R)) . x by A5, A6, A14, A11, FUNCOP_1:22 ; ::_thesis: verum end; hence lmlt ((p1 + p2),R) = (lmlt (p1,R)) + (lmlt (p2,R)) by A5, FUNCT_1:2; ::_thesis: verum end; theorem :: MATRLIN2:8 for K being Field for V1 being finite-dimensional VectSp of K for R1, R2 being FinSequence of V1 for p being FinSequence of K holds lmlt (p,(R1 + R2)) = (lmlt (p,R1)) + (lmlt (p,R2)) proof let K be Field; ::_thesis: for V1 being finite-dimensional VectSp of K for R1, R2 being FinSequence of V1 for p being FinSequence of K holds lmlt (p,(R1 + R2)) = (lmlt (p,R1)) + (lmlt (p,R2)) let V1 be finite-dimensional VectSp of K; ::_thesis: for R1, R2 being FinSequence of V1 for p being FinSequence of K holds lmlt (p,(R1 + R2)) = (lmlt (p,R1)) + (lmlt (p,R2)) let R1, R2 be FinSequence of V1; ::_thesis: for p being FinSequence of K holds lmlt (p,(R1 + R2)) = (lmlt (p,R1)) + (lmlt (p,R2)) let p be FinSequence of K; ::_thesis: lmlt (p,(R1 + R2)) = (lmlt (p,R1)) + (lmlt (p,R2)) set L12 = lmlt (p,(R1 + R2)); set L1 = lmlt (p,R1); set L2 = lmlt (p,R2); A1: dom ((lmlt (p,R1)) + (lmlt (p,R2))) = (dom (lmlt (p,R1))) /\ (dom (lmlt (p,R2))) by Lm3; A2: dom (lmlt (p,(R1 + R2))) = (dom p) /\ (dom (R1 + R2)) by Lm1; A3: dom (R1 + R2) = (dom R1) /\ (dom R2) by Lm3; A4: dom (lmlt (p,R1)) = (dom p) /\ (dom R1) by Lm1; A5: dom (lmlt (p,R2)) = (dom p) /\ (dom R2) by Lm1; then A6: dom ((lmlt (p,R1)) + (lmlt (p,R2))) = (((dom p) /\ (dom R1)) /\ (dom p)) /\ (dom R2) by A1, A4, XBOOLE_1:16 .= (((dom p) /\ (dom p)) /\ (dom R1)) /\ (dom R2) by XBOOLE_1:16 .= dom (lmlt (p,(R1 + R2))) by A3, A2, XBOOLE_1:16 ; now__::_thesis:_for_x_being_set_st_x_in_dom_((lmlt_(p,R1))_+_(lmlt_(p,R2)))_holds_ ((lmlt_(p,R1))_+_(lmlt_(p,R2)))_._x_=_(lmlt_(p,(R1_+_R2)))_._x let x be set ; ::_thesis: ( x in dom ((lmlt (p,R1)) + (lmlt (p,R2))) implies ((lmlt (p,R1)) + (lmlt (p,R2))) . x = (lmlt (p,(R1 + R2))) . x ) assume A7: x in dom ((lmlt (p,R1)) + (lmlt (p,R2))) ; ::_thesis: ((lmlt (p,R1)) + (lmlt (p,R2))) . x = (lmlt (p,(R1 + R2))) . x A8: x in dom (lmlt (p,R2)) by A1, A7, XBOOLE_0:def_4; then A9: (lmlt (p,R2)) /. x = (lmlt (p,R2)) . x by PARTFUN1:def_6; x in dom R2 by A5, A8, XBOOLE_0:def_4; then A10: R2 /. x = R2 . x by PARTFUN1:def_6; A11: x in dom (R1 + R2) by A2, A6, A7, XBOOLE_0:def_4; then A12: (R1 + R2) . x = (R1 + R2) /. x by PARTFUN1:def_6; A13: x in dom (lmlt (p,R1)) by A1, A7, XBOOLE_0:def_4; then x in dom p by A4, XBOOLE_0:def_4; then A14: p /. x = p . x by PARTFUN1:def_6; x in dom R1 by A4, A13, XBOOLE_0:def_4; then A15: R1 /. x = R1 . x by PARTFUN1:def_6; A16: (lmlt (p,R1)) /. x = (lmlt (p,R1)) . x by A13, PARTFUN1:def_6; hence ((lmlt (p,R1)) + (lmlt (p,R2))) . x = ((lmlt (p,R1)) /. x) + ((lmlt (p,R2)) /. x) by A7, A9, FVSUM_1:17 .= ( the lmult of V1 . ((p /. x),(R1 /. x))) + ((lmlt (p,R2)) /. x) by A13, A16, A14, A15, FUNCOP_1:22 .= ((p /. x) * (R1 /. x)) + ((p /. x) * (R2 /. x)) by A8, A9, A14, A10, FUNCOP_1:22 .= (p /. x) * ((R1 /. x) + (R2 /. x)) by VECTSP_1:def_14 .= (p /. x) * ((R1 + R2) /. x) by A11, A15, A10, A12, FVSUM_1:17 .= (lmlt (p,(R1 + R2))) . x by A6, A7, A14, A12, FUNCOP_1:22 ; ::_thesis: verum end; hence lmlt (p,(R1 + R2)) = (lmlt (p,R1)) + (lmlt (p,R2)) by A6, FUNCT_1:2; ::_thesis: verum end; theorem Th9: :: MATRLIN2:9 for K being Field for V1 being finite-dimensional VectSp of K for R1, R2 being FinSequence of V1 for p1, p2 being FinSequence of K st len p1 = len R1 & len p2 = len R2 holds lmlt ((p1 ^ p2),(R1 ^ R2)) = (lmlt (p1,R1)) ^ (lmlt (p2,R2)) proof let K be Field; ::_thesis: for V1 being finite-dimensional VectSp of K for R1, R2 being FinSequence of V1 for p1, p2 being FinSequence of K st len p1 = len R1 & len p2 = len R2 holds lmlt ((p1 ^ p2),(R1 ^ R2)) = (lmlt (p1,R1)) ^ (lmlt (p2,R2)) let V1 be finite-dimensional VectSp of K; ::_thesis: for R1, R2 being FinSequence of V1 for p1, p2 being FinSequence of K st len p1 = len R1 & len p2 = len R2 holds lmlt ((p1 ^ p2),(R1 ^ R2)) = (lmlt (p1,R1)) ^ (lmlt (p2,R2)) let R1, R2 be FinSequence of V1; ::_thesis: for p1, p2 being FinSequence of K st len p1 = len R1 & len p2 = len R2 holds lmlt ((p1 ^ p2),(R1 ^ R2)) = (lmlt (p1,R1)) ^ (lmlt (p2,R2)) let p1, p2 be FinSequence of K; ::_thesis: ( len p1 = len R1 & len p2 = len R2 implies lmlt ((p1 ^ p2),(R1 ^ R2)) = (lmlt (p1,R1)) ^ (lmlt (p2,R2)) ) assume that A1: len p1 = len R1 and A2: len p2 = len R2 ; ::_thesis: lmlt ((p1 ^ p2),(R1 ^ R2)) = (lmlt (p1,R1)) ^ (lmlt (p2,R2)) reconsider r2 = R2 as Element of (len p2) -tuples_on the carrier of V1 by A2, FINSEQ_2:92; reconsider r1 = R1 as Element of (len p1) -tuples_on the carrier of V1 by A1, FINSEQ_2:92; reconsider P1 = p1 as Element of (len p1) -tuples_on the carrier of K by FINSEQ_2:92; reconsider P2 = p2 as Element of (len p2) -tuples_on the carrier of K by FINSEQ_2:92; thus lmlt ((p1 ^ p2),(R1 ^ R2)) = ( the lmult of V1 .: (P1,r1)) ^ ( the lmult of V1 .: (P2,r2)) by FINSEQOP:11 .= (lmlt (p1,R1)) ^ (lmlt (p2,R2)) ; ::_thesis: verum end; theorem :: MATRLIN2:10 for K being Field for V1 being finite-dimensional VectSp of K for R1, R2 being FinSequence of V1 st len R1 = len R2 holds Sum (R1 + R2) = (Sum R1) + (Sum R2) proof let K be Field; ::_thesis: for V1 being finite-dimensional VectSp of K for R1, R2 being FinSequence of V1 st len R1 = len R2 holds Sum (R1 + R2) = (Sum R1) + (Sum R2) let V1 be finite-dimensional VectSp of K; ::_thesis: for R1, R2 being FinSequence of V1 st len R1 = len R2 holds Sum (R1 + R2) = (Sum R1) + (Sum R2) let R1, R2 be FinSequence of V1; ::_thesis: ( len R1 = len R2 implies Sum (R1 + R2) = (Sum R1) + (Sum R2) ) assume len R1 = len R2 ; ::_thesis: Sum (R1 + R2) = (Sum R1) + (Sum R2) then reconsider r1 = R1, r2 = R2 as Element of (len R1) -tuples_on the carrier of V1 by FINSEQ_2:92; thus Sum (R1 + R2) = Sum (r1 + r2) .= (Sum R1) + (Sum R2) by FVSUM_1:76 ; ::_thesis: verum end; theorem :: MATRLIN2:11 for K being Field for a being Element of K for V1 being finite-dimensional VectSp of K for R being FinSequence of V1 holds Sum (lmlt (((len R) |-> a),R)) = a * (Sum R) proof let K be Field; ::_thesis: for a being Element of K for V1 being finite-dimensional VectSp of K for R being FinSequence of V1 holds Sum (lmlt (((len R) |-> a),R)) = a * (Sum R) let a be Element of K; ::_thesis: for V1 being finite-dimensional VectSp of K for R being FinSequence of V1 holds Sum (lmlt (((len R) |-> a),R)) = a * (Sum R) let V1 be finite-dimensional VectSp of K; ::_thesis: for R being FinSequence of V1 holds Sum (lmlt (((len R) |-> a),R)) = a * (Sum R) let R be FinSequence of V1; ::_thesis: Sum (lmlt (((len R) |-> a),R)) = a * (Sum R) defpred S1[ Nat] means for R being FinSequence of V1 for a being Element of K st len R = $1 holds Sum (lmlt (((len R) |-> a),R)) = a * (Sum R); A1: for n being Nat st S1[n] holds S1[n + 1] proof let n be Nat; ::_thesis: ( S1[n] implies S1[n + 1] ) assume A2: S1[n] ; ::_thesis: S1[n + 1] set n1 = n + 1; let R be FinSequence of V1; ::_thesis: for a being Element of K st len R = n + 1 holds Sum (lmlt (((len R) |-> a),R)) = a * (Sum R) let a be Element of K; ::_thesis: ( len R = n + 1 implies Sum (lmlt (((len R) |-> a),R)) = a * (Sum R) ) assume A3: len R = n + 1 ; ::_thesis: Sum (lmlt (((len R) |-> a),R)) = a * (Sum R) A4: len (R | n) = n by A3, FINSEQ_1:59, NAT_1:11; then A5: dom (R | n) = Seg n by FINSEQ_1:def_3; 1 <= n + 1 by NAT_1:11; then n + 1 in dom R by A3, FINSEQ_3:25; then A6: R /. (n + 1) = R . (n + 1) by PARTFUN1:def_6; A7: lmlt (<*a*>,<*(R /. (n + 1))*>) = <*(a * (R /. (n + 1)))*> by FINSEQ_2:74; A8: ( len <*a*> = 1 & len <*(R . (n + 1))*> = 1 ) by FINSEQ_1:39; A9: ( (n + 1) |-> a = (n |-> a) ^ <*a*> & len (n |-> a) = n ) by CARD_1:def_7, FINSEQ_2:60; R = (R | n) ^ <*(R . (n + 1))*> by A3, FINSEQ_3:55; hence Sum (lmlt (((len R) |-> a),R)) = Sum ((lmlt ((n |-> a),(R | n))) ^ (lmlt (<*a*>,<*(R /. (n + 1))*>))) by A3, A6, A4, A9, A8, Th9 .= (Sum (lmlt ((n |-> a),(R | n)))) + (Sum (lmlt (<*a*>,<*(R /. (n + 1))*>))) by RLVECT_1:41 .= (a * (Sum (R | n))) + (Sum <*(a * (R /. (n + 1)))*>) by A2, A4, A7 .= (a * (Sum (R | n))) + (a * (R /. (n + 1))) by RLVECT_1:44 .= a * ((Sum (R | n)) + (R /. (n + 1))) by VECTSP_1:def_14 .= a * (Sum R) by A3, A6, A4, A5, RLVECT_1:38 ; ::_thesis: verum end; A10: S1[ 0 ] proof let R be FinSequence of V1; ::_thesis: for a being Element of K st len R = 0 holds Sum (lmlt (((len R) |-> a),R)) = a * (Sum R) let a be Element of K; ::_thesis: ( len R = 0 implies Sum (lmlt (((len R) |-> a),R)) = a * (Sum R) ) assume A11: len R = 0 ; ::_thesis: Sum (lmlt (((len R) |-> a),R)) = a * (Sum R) set L = (len R) |-> a; set M = lmlt (((len R) |-> a),R); len ((len R) |-> a) = len R by CARD_1:def_7; then dom ((len R) |-> a) = dom R by FINSEQ_3:29; then dom (lmlt (((len R) |-> a),R)) = dom R by MATRLIN:12; then len R = len (lmlt (((len R) |-> a),R)) by FINSEQ_3:29; then lmlt (((len R) |-> a),R) = <*> the carrier of V1 by A11; then A12: Sum (lmlt (((len R) |-> a),R)) = 0. V1 by RLVECT_1:43; R = <*> the carrier of V1 by A11; then Sum R = 0. V1 by RLVECT_1:43; hence Sum (lmlt (((len R) |-> a),R)) = a * (Sum R) by A12, VECTSP_1:14; ::_thesis: verum end; for n being Nat holds S1[n] from NAT_1:sch_2(A10, A1); hence Sum (lmlt (((len R) |-> a),R)) = a * (Sum R) ; ::_thesis: verum end; theorem :: MATRLIN2:12 for K being Field for V1 being finite-dimensional VectSp of K for v1 being Element of V1 for p being FinSequence of K holds Sum (lmlt (p,((len p) |-> v1))) = (Sum p) * v1 proof let K be Field; ::_thesis: for V1 being finite-dimensional VectSp of K for v1 being Element of V1 for p being FinSequence of K holds Sum (lmlt (p,((len p) |-> v1))) = (Sum p) * v1 let V1 be finite-dimensional VectSp of K; ::_thesis: for v1 being Element of V1 for p being FinSequence of K holds Sum (lmlt (p,((len p) |-> v1))) = (Sum p) * v1 let v1 be Element of V1; ::_thesis: for p being FinSequence of K holds Sum (lmlt (p,((len p) |-> v1))) = (Sum p) * v1 let p be FinSequence of K; ::_thesis: Sum (lmlt (p,((len p) |-> v1))) = (Sum p) * v1 set L = (len p) |-> v1; set M = lmlt (p,((len p) |-> v1)); len ((len p) |-> v1) = len p by CARD_1:def_7; then dom ((len p) |-> v1) = dom p by FINSEQ_3:29; then A1: dom (lmlt (p,((len p) |-> v1))) = dom p by MATRLIN:12; A2: now__::_thesis:_for_k_being_Nat for_a1_being_Element_of_K_st_k_in_dom_(lmlt_(p,((len_p)_|->_v1)))_&_a1_=_p_._k_holds_ (lmlt_(p,((len_p)_|->_v1)))_._k_=_a1_*_v1 let k be Nat; ::_thesis: for a1 being Element of K st k in dom (lmlt (p,((len p) |-> v1))) & a1 = p . k holds (lmlt (p,((len p) |-> v1))) . k = a1 * v1 let a1 be Element of K; ::_thesis: ( k in dom (lmlt (p,((len p) |-> v1))) & a1 = p . k implies (lmlt (p,((len p) |-> v1))) . k = a1 * v1 ) assume that A3: k in dom (lmlt (p,((len p) |-> v1))) and A4: a1 = p . k ; ::_thesis: (lmlt (p,((len p) |-> v1))) . k = a1 * v1 k in Seg (len p) by A1, A3, FINSEQ_1:def_3; then ((len p) |-> v1) . k = v1 by FINSEQ_2:57; hence (lmlt (p,((len p) |-> v1))) . k = a1 * v1 by A3, A4, FUNCOP_1:22; ::_thesis: verum end; len p = len (lmlt (p,((len p) |-> v1))) by A1, FINSEQ_3:29; hence Sum (lmlt (p,((len p) |-> v1))) = (Sum p) * v1 by A2, MATRLIN:9; ::_thesis: verum end; theorem :: MATRLIN2:13 for K being Field for a being Element of K for V1 being finite-dimensional VectSp of K for R being FinSequence of V1 for p being FinSequence of K holds Sum (lmlt ((a * p),R)) = a * (Sum (lmlt (p,R))) proof let K be Field; ::_thesis: for a being Element of K for V1 being finite-dimensional VectSp of K for R being FinSequence of V1 for p being FinSequence of K holds Sum (lmlt ((a * p),R)) = a * (Sum (lmlt (p,R))) let a be Element of K; ::_thesis: for V1 being finite-dimensional VectSp of K for R being FinSequence of V1 for p being FinSequence of K holds Sum (lmlt ((a * p),R)) = a * (Sum (lmlt (p,R))) let V1 be finite-dimensional VectSp of K; ::_thesis: for R being FinSequence of V1 for p being FinSequence of K holds Sum (lmlt ((a * p),R)) = a * (Sum (lmlt (p,R))) let R be FinSequence of V1; ::_thesis: for p being FinSequence of K holds Sum (lmlt ((a * p),R)) = a * (Sum (lmlt (p,R))) let p be FinSequence of K; ::_thesis: Sum (lmlt ((a * p),R)) = a * (Sum (lmlt (p,R))) set Ma = lmlt ((a * p),R); set M = lmlt (p,R); len (a * p) = len p by MATRIXR1:16; then A1: dom (a * p) = dom p by FINSEQ_3:29; A2: dom (lmlt ((a * p),R)) = (dom (a * p)) /\ (dom R) by Lm1; A3: dom (lmlt (p,R)) = (dom p) /\ (dom R) by Lm1; A4: for k being Element of NAT for v1 being Element of V1 st k in dom (lmlt ((a * p),R)) & v1 = (lmlt (p,R)) . k holds (lmlt ((a * p),R)) . k = a * v1 proof let k be Element of NAT ; ::_thesis: for v1 being Element of V1 st k in dom (lmlt ((a * p),R)) & v1 = (lmlt (p,R)) . k holds (lmlt ((a * p),R)) . k = a * v1 let v1 be Element of V1; ::_thesis: ( k in dom (lmlt ((a * p),R)) & v1 = (lmlt (p,R)) . k implies (lmlt ((a * p),R)) . k = a * v1 ) assume that A5: k in dom (lmlt ((a * p),R)) and A6: v1 = (lmlt (p,R)) . k ; ::_thesis: (lmlt ((a * p),R)) . k = a * v1 k in dom R by A2, A5, XBOOLE_0:def_4; then A7: R /. k = R . k by PARTFUN1:def_6; k in dom p by A1, A2, A5, XBOOLE_0:def_4; then A8: p /. k = p . k by PARTFUN1:def_6; k in dom (a * p) by A2, A5, XBOOLE_0:def_4; then (a * p) . k = a * (p /. k) by A8, FVSUM_1:50; hence (lmlt ((a * p),R)) . k = (a * (p /. k)) * (R /. k) by A5, A7, FUNCOP_1:22 .= a * ((p /. k) * (R /. k)) by VECTSP_1:def_16 .= a * v1 by A1, A3, A2, A5, A6, A8, A7, FUNCOP_1:22 ; ::_thesis: verum end; len (lmlt (p,R)) = len (lmlt ((a * p),R)) by A1, A3, A2, FINSEQ_3:29; hence Sum (lmlt ((a * p),R)) = a * (Sum (lmlt (p,R))) by A4, RLVECT_2:66; ::_thesis: verum end; theorem :: MATRLIN2:14 for K being Field for V1 being finite-dimensional VectSp of K for p being FinSequence of K for B1 being FinSequence of V1 for W1 being Subspace of V1 for B2 being FinSequence of W1 st B1 = B2 holds lmlt (p,B1) = lmlt (p,B2) proof let K be Field; ::_thesis: for V1 being finite-dimensional VectSp of K for p being FinSequence of K for B1 being FinSequence of V1 for W1 being Subspace of V1 for B2 being FinSequence of W1 st B1 = B2 holds lmlt (p,B1) = lmlt (p,B2) let V1 be finite-dimensional VectSp of K; ::_thesis: for p being FinSequence of K for B1 being FinSequence of V1 for W1 being Subspace of V1 for B2 being FinSequence of W1 st B1 = B2 holds lmlt (p,B1) = lmlt (p,B2) let p be FinSequence of K; ::_thesis: for B1 being FinSequence of V1 for W1 being Subspace of V1 for B2 being FinSequence of W1 st B1 = B2 holds lmlt (p,B1) = lmlt (p,B2) let B1 be FinSequence of V1; ::_thesis: for W1 being Subspace of V1 for B2 being FinSequence of W1 st B1 = B2 holds lmlt (p,B1) = lmlt (p,B2) let W1 be Subspace of V1; ::_thesis: for B2 being FinSequence of W1 st B1 = B2 holds lmlt (p,B1) = lmlt (p,B2) let B2 be FinSequence of W1; ::_thesis: ( B1 = B2 implies lmlt (p,B1) = lmlt (p,B2) ) assume A1: B1 = B2 ; ::_thesis: lmlt (p,B1) = lmlt (p,B2) set M2 = lmlt (p,B2); set M1 = lmlt (p,B1); A2: dom (lmlt (p,B1)) = (dom p) /\ (dom B1) by Lm1; A3: dom (lmlt (p,B2)) = (dom p) /\ (dom B2) by Lm1; now__::_thesis:_for_i_being_Nat_st_i_in_dom_(lmlt_(p,B1))_holds_ (lmlt_(p,B1))_._i_=_(lmlt_(p,B2))_._i let i be Nat; ::_thesis: ( i in dom (lmlt (p,B1)) implies (lmlt (p,B1)) . i = (lmlt (p,B2)) . i ) assume A4: i in dom (lmlt (p,B1)) ; ::_thesis: (lmlt (p,B1)) . i = (lmlt (p,B2)) . i i in dom p by A2, A4, XBOOLE_0:def_4; then A5: p . i = p /. i by PARTFUN1:def_6; A6: i in dom B1 by A2, A4, XBOOLE_0:def_4; then A7: B2 . i = B2 /. i by A1, PARTFUN1:def_6; A8: B1 . i = B1 /. i by A6, PARTFUN1:def_6; hence (lmlt (p,B1)) . i = (p /. i) * (B1 /. i) by A4, A5, FUNCOP_1:22 .= (p /. i) * (B2 /. i) by A1, A6, A8, PARTFUN1:def_6, VECTSP_4:14 .= (lmlt (p,B2)) . i by A1, A2, A3, A4, A5, A7, FUNCOP_1:22 ; ::_thesis: verum end; hence lmlt (p,B1) = lmlt (p,B2) by A1, A3, Lm1, FINSEQ_1:13; ::_thesis: verum end; theorem :: MATRLIN2:15 for K being Field for V1 being finite-dimensional VectSp of K for B1 being FinSequence of V1 for W1 being Subspace of V1 for B2 being FinSequence of W1 st B1 = B2 holds Sum B1 = Sum B2 proof let K be Field; ::_thesis: for V1 being finite-dimensional VectSp of K for B1 being FinSequence of V1 for W1 being Subspace of V1 for B2 being FinSequence of W1 st B1 = B2 holds Sum B1 = Sum B2 let V1 be finite-dimensional VectSp of K; ::_thesis: for B1 being FinSequence of V1 for W1 being Subspace of V1 for B2 being FinSequence of W1 st B1 = B2 holds Sum B1 = Sum B2 let B1 be FinSequence of V1; ::_thesis: for W1 being Subspace of V1 for B2 being FinSequence of W1 st B1 = B2 holds Sum B1 = Sum B2 let W1 be Subspace of V1; ::_thesis: for B2 being FinSequence of W1 st B1 = B2 holds Sum B1 = Sum B2 let B2 be FinSequence of W1; ::_thesis: ( B1 = B2 implies Sum B1 = Sum B2 ) assume A1: B1 = B2 ; ::_thesis: Sum B1 = Sum B2 defpred S1[ Nat] means for B1 being FinSequence of V1 for W1 being Subspace of V1 for B2 being FinSequence of W1 st B1 = B2 & len B1 = $1 holds Sum B1 = Sum B2; A2: for n being Nat st S1[n] holds S1[n + 1] proof let n be Nat; ::_thesis: ( S1[n] implies S1[n + 1] ) assume A3: S1[n] ; ::_thesis: S1[n + 1] set n1 = n + 1; let B1 be FinSequence of V1; ::_thesis: for W1 being Subspace of V1 for B2 being FinSequence of W1 st B1 = B2 & len B1 = n + 1 holds Sum B1 = Sum B2 let W1 be Subspace of V1; ::_thesis: for B2 being FinSequence of W1 st B1 = B2 & len B1 = n + 1 holds Sum B1 = Sum B2 let B2 be FinSequence of W1; ::_thesis: ( B1 = B2 & len B1 = n + 1 implies Sum B1 = Sum B2 ) assume that A4: B1 = B2 and A5: len B1 = n + 1 ; ::_thesis: Sum B1 = Sum B2 A6: len (B1 | n) = n by A5, FINSEQ_1:59, NAT_1:11; then A7: Sum (B1 | n) = Sum (B2 | n) by A3, A4; 1 <= n + 1 by NAT_1:11; then A8: n + 1 in dom B1 by A5, FINSEQ_3:25; then A9: B2 . (n + 1) = B2 /. (n + 1) by A4, PARTFUN1:def_6; A10: B1 . (n + 1) = B1 /. (n + 1) by A8, PARTFUN1:def_6; A11: dom (B1 | n) = Seg n by A6, FINSEQ_1:def_3; hence Sum B1 = (Sum (B1 | n)) + (B1 /. (n + 1)) by A5, A10, A6, RLVECT_1:38 .= (Sum (B2 | n)) + (B2 /. (n + 1)) by A4, A10, A9, A7, VECTSP_4:13 .= Sum B2 by A4, A5, A9, A6, A11, RLVECT_1:38 ; ::_thesis: verum end; A12: S1[ 0 ] proof let B1 be FinSequence of V1; ::_thesis: for W1 being Subspace of V1 for B2 being FinSequence of W1 st B1 = B2 & len B1 = 0 holds Sum B1 = Sum B2 let W1 be Subspace of V1; ::_thesis: for B2 being FinSequence of W1 st B1 = B2 & len B1 = 0 holds Sum B1 = Sum B2 let B2 be FinSequence of W1; ::_thesis: ( B1 = B2 & len B1 = 0 implies Sum B1 = Sum B2 ) assume ( B1 = B2 & len B1 = 0 ) ; ::_thesis: Sum B1 = Sum B2 then ( Sum B1 = 0. V1 & Sum B2 = 0. W1 ) by RLVECT_1:75; hence Sum B1 = Sum B2 by VECTSP_4:11; ::_thesis: verum end; for n being Nat holds S1[n] from NAT_1:sch_2(A12, A2); then S1[ len B1] ; hence Sum B1 = Sum B2 by A1; ::_thesis: verum end; theorem :: MATRLIN2:16 for i being Nat for K being Field for V1 being finite-dimensional VectSp of K for R being FinSequence of V1 st i in dom R holds Sum (lmlt ((Line ((1. (K,(len R))),i)),R)) = R /. i proof let i be Nat; ::_thesis: for K being Field for V1 being finite-dimensional VectSp of K for R being FinSequence of V1 st i in dom R holds Sum (lmlt ((Line ((1. (K,(len R))),i)),R)) = R /. i let K be Field; ::_thesis: for V1 being finite-dimensional VectSp of K for R being FinSequence of V1 st i in dom R holds Sum (lmlt ((Line ((1. (K,(len R))),i)),R)) = R /. i let V1 be finite-dimensional VectSp of K; ::_thesis: for R being FinSequence of V1 st i in dom R holds Sum (lmlt ((Line ((1. (K,(len R))),i)),R)) = R /. i let R be FinSequence of V1; ::_thesis: ( i in dom R implies Sum (lmlt ((Line ((1. (K,(len R))),i)),R)) = R /. i ) set ONE = 1. (K,(len R)); set L = Line ((1. (K,(len R))),i); set M = lmlt ((Line ((1. (K,(len R))),i)),R); A1: width (1. (K,(len R))) = len R by MATRIX_1:24; len (Line ((1. (K,(len R))),i)) = width (1. (K,(len R))) by CARD_1:def_7; then dom (Line ((1. (K,(len R))),i)) = dom R by A1, FINSEQ_3:29; then A2: dom (lmlt ((Line ((1. (K,(len R))),i)),R)) = dom R by MATRLIN:12; then A3: len (lmlt ((Line ((1. (K,(len R))),i)),R)) = len R by FINSEQ_3:29; consider f being Function of NAT, the carrier of V1 such that A4: Sum (lmlt ((Line ((1. (K,(len R))),i)),R)) = f . (len (lmlt ((Line ((1. (K,(len R))),i)),R))) and A5: f . 0 = 0. V1 and A6: for j being Element of NAT for v1 being Element of V1 st j < len (lmlt ((Line ((1. (K,(len R))),i)),R)) & v1 = (lmlt ((Line ((1. (K,(len R))),i)),R)) . (j + 1) holds f . (j + 1) = (f . j) + v1 by RLVECT_1:def_12; defpred S1[ Nat] means ( $1 <= len (lmlt ((Line ((1. (K,(len R))),i)),R)) implies f . $1 = R /. i ); defpred S2[ Nat] means ( $1 < i implies f . $1 = 0. V1 ); assume A7: i in dom R ; ::_thesis: Sum (lmlt ((Line ((1. (K,(len R))),i)),R)) = R /. i then A8: 1 <= i by FINSEQ_3:25; len (1. (K,(len R))) = len R by MATRIX_1:24; then A9: dom R = dom (1. (K,(len R))) by FINSEQ_3:29; A10: for n being Nat st i <= n & S1[n] holds S1[n + 1] proof let n be Nat; ::_thesis: ( i <= n & S1[n] implies S1[n + 1] ) assume A11: i <= n ; ::_thesis: ( not S1[n] or S1[n + 1] ) set n1 = n + 1; A12: i < n + 1 by A11, NAT_1:13; reconsider N = n as Element of NAT by ORDINAL1:def_12; assume A13: S1[n] ; ::_thesis: S1[n + 1] assume A14: n + 1 <= len (lmlt ((Line ((1. (K,(len R))),i)),R)) ; ::_thesis: f . (n + 1) = R /. i then A15: n < len (lmlt ((Line ((1. (K,(len R))),i)),R)) by NAT_1:13; A16: 1 <= n + 1 by NAT_1:11; then n + 1 in Seg (len R) by A3, A14; then ( (Line ((1. (K,(len R))),i)) . (n + 1) = (1. (K,(len R))) * (i,(n + 1)) & [i,(n + 1)] in Indices (1. (K,(len R))) ) by A7, A1, A9, MATRIX_1:def_7, ZFMISC_1:87; then A17: (Line ((1. (K,(len R))),i)) . (n + 1) = 0. K by A12, MATRIX_1:def_11; A18: n + 1 in dom R by A2, A14, A16, FINSEQ_3:25; then R . (n + 1) = R /. (n + 1) by PARTFUN1:def_6; then (lmlt ((Line ((1. (K,(len R))),i)),R)) . (n + 1) = (0. K) * (R /. (n + 1)) by A2, A18, A17, FUNCOP_1:22 .= 0. V1 by VECTSP_1:14 ; hence f . (n + 1) = (f . N) + (0. V1) by A6, A15 .= R /. i by A13, A14, NAT_1:13, RLVECT_1:def_4 ; ::_thesis: verum end; A19: i <= len (lmlt ((Line ((1. (K,(len R))),i)),R)) by A7, A2, FINSEQ_3:25; A20: for n being Nat st S2[n] holds S2[n + 1] proof let n be Nat; ::_thesis: ( S2[n] implies S2[n + 1] ) assume A21: S2[n] ; ::_thesis: S2[n + 1] reconsider N = n as Element of NAT by ORDINAL1:def_12; set n1 = n + 1; assume A22: n + 1 < i ; ::_thesis: f . (n + 1) = 0. V1 then n + 1 < len (lmlt ((Line ((1. (K,(len R))),i)),R)) by A19, XXREAL_0:2; then A23: n < len (lmlt ((Line ((1. (K,(len R))),i)),R)) by NAT_1:13; A24: ( 1 <= n + 1 & n + 1 <= len R ) by A19, A3, A22, NAT_1:11, XXREAL_0:2; then n + 1 in Seg (len R) ; then ( (Line ((1. (K,(len R))),i)) . (n + 1) = (1. (K,(len R))) * (i,(n + 1)) & [i,(n + 1)] in Indices (1. (K,(len R))) ) by A7, A1, A9, MATRIX_1:def_7, ZFMISC_1:87; then A25: (Line ((1. (K,(len R))),i)) . (n + 1) = 0. K by A22, MATRIX_1:def_11; A26: n + 1 in dom R by A24, FINSEQ_3:25; then R . (n + 1) = R /. (n + 1) by PARTFUN1:def_6; then (lmlt ((Line ((1. (K,(len R))),i)),R)) . (n + 1) = (0. K) * (R /. (n + 1)) by A2, A26, A25, FUNCOP_1:22 .= 0. V1 by VECTSP_1:14 ; hence f . (n + 1) = (f . N) + (0. V1) by A6, A23 .= 0. V1 by A21, A22, NAT_1:13, RLVECT_1:def_4 ; ::_thesis: verum end; A27: S2[ 0 ] by A5; A28: for n being Nat holds S2[n] from NAT_1:sch_2(A27, A20); A29: S1[i] proof i in Seg (len R) by A7, A8, A19, A3; then ( (Line ((1. (K,(len R))),i)) . i = (1. (K,(len R))) * (i,i) & [i,i] in Indices (1. (K,(len R))) ) by A7, A1, A9, MATRIX_1:def_7, ZFMISC_1:87; then A30: (Line ((1. (K,(len R))),i)) . i = 1_ K by MATRIX_1:def_11; reconsider i1 = i - 1 as Element of NAT by A8, NAT_1:21; A31: i1 + 1 = i ; then i1 < i by NAT_1:13; then A32: f . i1 = 0. V1 by A28; assume i <= len (lmlt ((Line ((1. (K,(len R))),i)),R)) ; ::_thesis: f . i = R /. i then A33: i1 < len (lmlt ((Line ((1. (K,(len R))),i)),R)) by A31, NAT_1:13; R . i = R /. i by A7, PARTFUN1:def_6; then (lmlt ((Line ((1. (K,(len R))),i)),R)) . i = (1_ K) * (R /. i) by A7, A2, A30, FUNCOP_1:22 .= R /. i by VECTSP_1:def_17 ; then f . (i1 + 1) = (f . i1) + (R /. i) by A6, A33; hence f . i = R /. i by A32, RLVECT_1:def_4; ::_thesis: verum end; for n being Nat st i <= n holds S1[n] from NAT_1:sch_8(A29, A10); hence Sum (lmlt ((Line ((1. (K,(len R))),i)),R)) = R /. i by A19, A4; ::_thesis: verum end; begin theorem Th17: :: MATRLIN2:17 for K being Field for V1 being finite-dimensional VectSp of K for b1 being OrdBasis of V1 for v1, w1 being Element of V1 holds (v1 + w1) |-- b1 = (v1 |-- b1) + (w1 |-- b1) proof let K be Field; ::_thesis: for V1 being finite-dimensional VectSp of K for b1 being OrdBasis of V1 for v1, w1 being Element of V1 holds (v1 + w1) |-- b1 = (v1 |-- b1) + (w1 |-- b1) let V1 be finite-dimensional VectSp of K; ::_thesis: for b1 being OrdBasis of V1 for v1, w1 being Element of V1 holds (v1 + w1) |-- b1 = (v1 |-- b1) + (w1 |-- b1) let b1 be OrdBasis of V1; ::_thesis: for v1, w1 being Element of V1 holds (v1 + w1) |-- b1 = (v1 |-- b1) + (w1 |-- b1) let v1, w1 be Element of V1; ::_thesis: (v1 + w1) |-- b1 = (v1 |-- b1) + (w1 |-- b1) set vb = v1 |-- b1; set wb = w1 |-- b1; set vwb = (v1 + w1) |-- b1; consider L1 being Linear_Combination of V1 such that A1: ( v1 = Sum L1 & Carrier L1 c= rng b1 ) and A2: for k being Nat st 1 <= k & k <= len (v1 |-- b1) holds (v1 |-- b1) /. k = L1 . (b1 /. k) by MATRLIN:def_7; consider L3 being Linear_Combination of V1 such that A3: ( v1 + w1 = Sum L3 & Carrier L3 c= rng b1 ) and A4: for k being Nat st 1 <= k & k <= len ((v1 + w1) |-- b1) holds ((v1 + w1) |-- b1) /. k = L3 . (b1 /. k) by MATRLIN:def_7; A5: len (w1 |-- b1) = len b1 by MATRLIN:def_7; reconsider rb1 = rng b1 as Basis of V1 by MATRLIN:def_2; consider L2 being Linear_Combination of V1 such that A6: ( w1 = Sum L2 & Carrier L2 c= rng b1 ) and A7: for k being Nat st 1 <= k & k <= len (w1 |-- b1) holds (w1 |-- b1) /. k = L2 . (b1 /. k) by MATRLIN:def_7; A8: len (v1 |-- b1) = len b1 by MATRLIN:def_7; A9: len ((v1 + w1) |-- b1) = len b1 by MATRLIN:def_7; then reconsider vb = v1 |-- b1, wb = w1 |-- b1, vwb = (v1 + w1) |-- b1 as Element of (len b1) -tuples_on the carrier of K by A8, A5, FINSEQ_2:92; rb1 is linearly-independent by VECTSP_7:def_3; then A10: L3 = L1 + L2 by A1, A6, A3, MATRLIN:6; now__::_thesis:_for_i_being_Nat_st_i_in_Seg_(len_b1)_holds_ vwb_._i_=_(vb_+_wb)_._i A11: dom b1 = Seg (len b1) by FINSEQ_1:def_3; let i be Nat; ::_thesis: ( i in Seg (len b1) implies vwb . i = (vb + wb) . i ) assume A12: i in Seg (len b1) ; ::_thesis: vwb . i = (vb + wb) . i A13: ( 1 <= i & i <= len b1 ) by A12, FINSEQ_1:1; dom wb = dom b1 by A5, FINSEQ_3:29; then A14: wb . i = wb /. i by A12, A11, PARTFUN1:def_6; dom vb = dom b1 by A8, FINSEQ_3:29; then A15: vb . i = vb /. i by A12, A11, PARTFUN1:def_6; dom vwb = dom b1 by A9, FINSEQ_3:29; then vwb . i = vwb /. i by A12, A11, PARTFUN1:def_6; hence vwb . i = (L1 + L2) . (b1 /. i) by A4, A9, A10, A13 .= (L1 . (b1 /. i)) + (L2 . (b1 /. i)) by VECTSP_6:22 .= (vb /. i) + (L2 . (b1 /. i)) by A2, A8, A13 .= (vb /. i) + (wb /. i) by A7, A5, A13 .= (vb + wb) . i by A12, A15, A14, FVSUM_1:18 ; ::_thesis: verum end; hence (v1 + w1) |-- b1 = (v1 |-- b1) + (w1 |-- b1) by FINSEQ_2:119; ::_thesis: verum end; theorem Th18: :: MATRLIN2:18 for K being Field for a being Element of K for V1 being finite-dimensional VectSp of K for b1 being OrdBasis of V1 for v1 being Element of V1 holds (a * v1) |-- b1 = a * (v1 |-- b1) proof let K be Field; ::_thesis: for a being Element of K for V1 being finite-dimensional VectSp of K for b1 being OrdBasis of V1 for v1 being Element of V1 holds (a * v1) |-- b1 = a * (v1 |-- b1) let a be Element of K; ::_thesis: for V1 being finite-dimensional VectSp of K for b1 being OrdBasis of V1 for v1 being Element of V1 holds (a * v1) |-- b1 = a * (v1 |-- b1) let V1 be finite-dimensional VectSp of K; ::_thesis: for b1 being OrdBasis of V1 for v1 being Element of V1 holds (a * v1) |-- b1 = a * (v1 |-- b1) let b1 be OrdBasis of V1; ::_thesis: for v1 being Element of V1 holds (a * v1) |-- b1 = a * (v1 |-- b1) let v1 be Element of V1; ::_thesis: (a * v1) |-- b1 = a * (v1 |-- b1) set vb = v1 |-- b1; set avb = (a * v1) |-- b1; consider L1 being Linear_Combination of V1 such that A1: ( v1 = Sum L1 & Carrier L1 c= rng b1 ) and A2: for k being Nat st 1 <= k & k <= len (v1 |-- b1) holds (v1 |-- b1) /. k = L1 . (b1 /. k) by MATRLIN:def_7; A3: len (v1 |-- b1) = len b1 by MATRLIN:def_7; reconsider rb1 = rng b1 as Basis of V1 by MATRLIN:def_2; consider L2 being Linear_Combination of V1 such that A4: a * v1 = Sum L2 and A5: Carrier L2 c= rng b1 and A6: for k being Nat st 1 <= k & k <= len ((a * v1) |-- b1) holds ((a * v1) |-- b1) /. k = L2 . (b1 /. k) by MATRLIN:def_7; A7: len ((a * v1) |-- b1) = len b1 by MATRLIN:def_7; len (a * (v1 |-- b1)) = len (v1 |-- b1) by MATRIXR1:16; then reconsider vb9 = v1 |-- b1, avb = (a * v1) |-- b1, Avb = a * (v1 |-- b1) as Element of (len b1) -tuples_on the carrier of K by A3, A7, FINSEQ_2:92; A8: rb1 is linearly-independent by VECTSP_7:def_3; now__::_thesis:_for_i_being_Nat_st_i_in_Seg_(len_b1)_holds_ avb_._i_=_Avb_._i let i be Nat; ::_thesis: ( i in Seg (len b1) implies avb . i = Avb . i ) assume A9: i in Seg (len b1) ; ::_thesis: avb . i = Avb . i A10: ( 1 <= i & i <= len b1 ) by A9, FINSEQ_1:1; A11: now__::_thesis:_L2_._(b1_/._i)_=_a_*_(vb9_/._i) percases ( a <> 0. K or a = 0. K ) ; suppose a <> 0. K ; ::_thesis: L2 . (b1 /. i) = a * (vb9 /. i) then a * L1 = L2 by A1, A4, A5, A8, MATRLIN:7; hence L2 . (b1 /. i) = a * (L1 . (b1 /. i)) by VECTSP_6:def_9 .= a * (vb9 /. i) by A2, A3, A10 ; ::_thesis: verum end; supposeA12: a = 0. K ; ::_thesis: L2 . (b1 /. i) = a * (vb9 /. i) then A13: a * v1 = 0. V1 by VECTSP_1:14; ( L2 is Linear_Combination of Carrier L2 & Carrier L2 is linearly-independent ) by A5, A8, VECTSP_6:7, VECTSP_7:1; then not b1 /. i in Carrier L2 by A4, A13, VECTSP_7:def_1; hence L2 . (b1 /. i) = 0. K .= a * (vb9 /. i) by A12, VECTSP_1:7 ; ::_thesis: verum end; end; end; A14: dom b1 = Seg (len b1) by FINSEQ_1:def_3; dom (v1 |-- b1) = dom b1 by A3, FINSEQ_3:29; then A15: (v1 |-- b1) . i = (v1 |-- b1) /. i by A9, A14, PARTFUN1:def_6; dom avb = dom b1 by A7, FINSEQ_3:29; then avb . i = avb /. i by A9, A14, PARTFUN1:def_6; hence avb . i = L2 . (b1 /. i) by A6, A7, A10 .= Avb . i by A9, A15, A11, FVSUM_1:51 ; ::_thesis: verum end; hence (a * v1) |-- b1 = a * (v1 |-- b1) by FINSEQ_2:119; ::_thesis: verum end; theorem Th19: :: MATRLIN2:19 for i being Nat for K being Field for V1 being finite-dimensional VectSp of K for b1 being OrdBasis of V1 st i in dom b1 holds (b1 /. i) |-- b1 = Line ((1. (K,(len b1))),i) proof let i be Nat; ::_thesis: for K being Field for V1 being finite-dimensional VectSp of K for b1 being OrdBasis of V1 st i in dom b1 holds (b1 /. i) |-- b1 = Line ((1. (K,(len b1))),i) let K be Field; ::_thesis: for V1 being finite-dimensional VectSp of K for b1 being OrdBasis of V1 st i in dom b1 holds (b1 /. i) |-- b1 = Line ((1. (K,(len b1))),i) let V1 be finite-dimensional VectSp of K; ::_thesis: for b1 being OrdBasis of V1 st i in dom b1 holds (b1 /. i) |-- b1 = Line ((1. (K,(len b1))),i) let b1 be OrdBasis of V1; ::_thesis: ( i in dom b1 implies (b1 /. i) |-- b1 = Line ((1. (K,(len b1))),i) ) set ONE = 1. (K,(len b1)); set bb = (b1 /. i) |-- b1; consider KL being Linear_Combination of V1 such that A1: ( b1 /. i = Sum KL & Carrier KL c= rng b1 ) and A2: for k being Nat st 1 <= k & k <= len ((b1 /. i) |-- b1) holds ((b1 /. i) |-- b1) /. k = KL . (b1 /. k) by MATRLIN:def_7; reconsider rb1 = rng b1 as Basis of V1 by MATRLIN:def_2; A3: rb1 is linearly-independent by VECTSP_7:def_3; b1 /. i in {(b1 /. i)} by TARSKI:def_1; then b1 /. i in Lin {(b1 /. i)} by VECTSP_7:8; then consider Lb being Linear_Combination of {(b1 /. i)} such that A4: b1 /. i = Sum Lb by VECTSP_7:7; assume A5: i in dom b1 ; ::_thesis: (b1 /. i) |-- b1 = Line ((1. (K,(len b1))),i) then A6: b1 . i = b1 /. i by PARTFUN1:def_6; then A7: Carrier Lb c= {(b1 . i)} by VECTSP_6:def_4; A8: b1 . i in rb1 by A5, FUNCT_1:def_3; then {(b1 . i)} c= rb1 by ZFMISC_1:31; then Carrier Lb c= rb1 by A7, XBOOLE_1:1; then A9: Lb = KL by A4, A1, A3, MATRLIN:5; A10: width (1. (K,(len b1))) = len b1 by MATRIX_1:24; A11: Indices (1. (K,(len b1))) = [:(Seg (len b1)),(Seg (len b1)):] by MATRIX_1:24; A12: len b1 = len ((b1 /. i) |-- b1) by MATRLIN:def_7; A13: b1 /. i <> 0. V1 by A6, A3, A8, VECTSP_7:2; A14: now__::_thesis:_for_j_being_Nat_st_1_<=_j_&_j_<=_len_((b1_/._i)_|--_b1)_holds_ (Line_((1._(K,(len_b1))),i))_._j_=_((b1_/._i)_|--_b1)_._j let j be Nat; ::_thesis: ( 1 <= j & j <= len ((b1 /. i) |-- b1) implies (Line ((1. (K,(len b1))),i)) . j = ((b1 /. i) |-- b1) . j ) assume A15: ( 1 <= j & j <= len ((b1 /. i) |-- b1) ) ; ::_thesis: (Line ((1. (K,(len b1))),i)) . j = ((b1 /. i) |-- b1) . j A16: j in Seg (len b1) by A12, A15, FINSEQ_1:1; i in Seg (len b1) by A5, FINSEQ_1:def_3; then A17: [i,j] in Indices (1. (K,(len b1))) by A11, A16, ZFMISC_1:87; A18: j in dom b1 by A12, A15, FINSEQ_3:25; A19: dom ((b1 /. i) |-- b1) = dom b1 by A12, FINSEQ_3:29; now__::_thesis:_(Line_((1._(K,(len_b1))),i))_._j_=_((b1_/._i)_|--_b1)_._j percases ( i = j or i <> j ) ; supposeA20: i = j ; ::_thesis: (Line ((1. (K,(len b1))),i)) . j = ((b1 /. i) |-- b1) . j (Lb . (b1 /. i)) * (b1 /. i) = b1 /. i by A4, VECTSP_6:17 .= (1_ K) * (b1 /. i) by VECTSP_1:def_17 ; then A21: 1_ K = KL . (b1 /. i) by A13, A9, VECTSP10:4 .= ((b1 /. i) |-- b1) /. j by A2, A15, A20 ; 1_ K = (1. (K,(len b1))) * (i,j) by A17, A20, MATRIX_1:def_11 .= (Line ((1. (K,(len b1))),i)) . j by A10, A16, MATRIX_1:def_7 ; hence (Line ((1. (K,(len b1))),i)) . j = ((b1 /. i) |-- b1) . j by A18, A19, A21, PARTFUN1:def_6; ::_thesis: verum end; supposeA22: i <> j ; ::_thesis: (Line ((1. (K,(len b1))),i)) . j = ((b1 /. i) |-- b1) . j b1 is one-to-one by MATRLIN:def_2; then b1 . i <> b1 . j by A5, A18, A22, FUNCT_1:def_4; then A23: not b1 . j in Carrier Lb by A7, TARSKI:def_1; A24: 0. K = (1. (K,(len b1))) * (i,j) by A17, A22, MATRIX_1:def_11 .= (Line ((1. (K,(len b1))),i)) . j by A10, A16, MATRIX_1:def_7 ; b1 . j = b1 /. j by A18, PARTFUN1:def_6; then 0. K = KL . (b1 /. j) by A9, A23 .= ((b1 /. i) |-- b1) /. j by A2, A15 ; hence (Line ((1. (K,(len b1))),i)) . j = ((b1 /. i) |-- b1) . j by A18, A19, A24, PARTFUN1:def_6; ::_thesis: verum end; end; end; hence (Line ((1. (K,(len b1))),i)) . j = ((b1 /. i) |-- b1) . j ; ::_thesis: verum end; len (Line ((1. (K,(len b1))),i)) = len b1 by A10, CARD_1:def_7; hence (b1 /. i) |-- b1 = Line ((1. (K,(len b1))),i) by A12, A14, FINSEQ_1:14; ::_thesis: verum end; theorem Th20: :: MATRLIN2:20 for K being Field for V1 being finite-dimensional VectSp of K for b1 being OrdBasis of V1 holds (0. V1) |-- b1 = (len b1) |-> (0. K) proof let K be Field; ::_thesis: for V1 being finite-dimensional VectSp of K for b1 being OrdBasis of V1 holds (0. V1) |-- b1 = (len b1) |-> (0. K) let V1 be finite-dimensional VectSp of K; ::_thesis: for b1 being OrdBasis of V1 holds (0. V1) |-- b1 = (len b1) |-> (0. K) let b1 be OrdBasis of V1; ::_thesis: (0. V1) |-- b1 = (len b1) |-> (0. K) percases ( dom b1 = {} or dom b1 <> {} ) ; supposeA1: dom b1 = {} ; ::_thesis: (0. V1) |-- b1 = (len b1) |-> (0. K) then A2: len b1 = 0 by CARD_1:27, RELAT_1:41; len ((0. V1) |-- b1) = len b1 by MATRLIN:def_7; hence (0. V1) |-- b1 = {} by A1, CARD_1:27, RELAT_1:41 .= (len b1) |-> (0. K) by A2 ; ::_thesis: verum end; suppose dom b1 <> {} ; ::_thesis: (0. V1) |-- b1 = (len b1) |-> (0. K) then consider x being set such that A3: x in dom b1 by XBOOLE_0:def_1; A4: width (1. (K,(len b1))) = len b1 by MATRIX_1:24; reconsider x = x as Nat by A3; 0. V1 = (b1 /. x) - (b1 /. x) by VECTSP_1:16 .= (b1 /. x) + ((- (1_ K)) * (b1 /. x)) by VECTSP_1:14 ; hence (0. V1) |-- b1 = ((b1 /. x) |-- b1) + (((- (1_ K)) * (b1 /. x)) |-- b1) by Th17 .= ((b1 /. x) |-- b1) + ((- (1_ K)) * ((b1 /. x) |-- b1)) by Th18 .= (Line ((1. (K,(len b1))),x)) + ((- (1_ K)) * ((b1 /. x) |-- b1)) by A3, Th19 .= (Line ((1. (K,(len b1))),x)) + ((- (1_ K)) * (Line ((1. (K,(len b1))),x))) by A3, Th19 .= (Line ((1. (K,(len b1))),x)) + (- (Line ((1. (K,(len b1))),x))) by FVSUM_1:59 .= (len b1) |-> (0. K) by A4, FVSUM_1:26 ; ::_thesis: verum end; end; end; theorem Th21: :: MATRLIN2:21 for K being Field for V1 being finite-dimensional VectSp of K for b1 being OrdBasis of V1 holds len b1 = dim V1 proof let K be Field; ::_thesis: for V1 being finite-dimensional VectSp of K for b1 being OrdBasis of V1 holds len b1 = dim V1 let V1 be finite-dimensional VectSp of K; ::_thesis: for b1 being OrdBasis of V1 holds len b1 = dim V1 let b1 be OrdBasis of V1; ::_thesis: len b1 = dim V1 reconsider R = rng b1 as Basis of V1 by MATRLIN:def_2; A1: b1 is one-to-one by MATRLIN:def_2; thus len b1 = card (Seg (len b1)) by FINSEQ_1:57 .= card (dom b1) by FINSEQ_1:def_3 .= card R by A1, CARD_1:70 .= dim V1 by VECTSP_9:def_1 ; ::_thesis: verum end; Lm4: for K being Field for V being VectSp of K for W1 being Subspace of V for L1 being Linear_Combination of W1 ex K1 being Linear_Combination of V st ( Carrier K1 = Carrier L1 & Sum K1 = Sum L1 & K1 | the carrier of W1 = L1 ) proof let K be Field; ::_thesis: for V being VectSp of K for W1 being Subspace of V for L1 being Linear_Combination of W1 ex K1 being Linear_Combination of V st ( Carrier K1 = Carrier L1 & Sum K1 = Sum L1 & K1 | the carrier of W1 = L1 ) let V be VectSp of K; ::_thesis: for W1 being Subspace of V for L1 being Linear_Combination of W1 ex K1 being Linear_Combination of V st ( Carrier K1 = Carrier L1 & Sum K1 = Sum L1 & K1 | the carrier of W1 = L1 ) let W1 be Subspace of V; ::_thesis: for L1 being Linear_Combination of W1 ex K1 being Linear_Combination of V st ( Carrier K1 = Carrier L1 & Sum K1 = Sum L1 & K1 | the carrier of W1 = L1 ) let L1 be Linear_Combination of W1; ::_thesis: ex K1 being Linear_Combination of V st ( Carrier K1 = Carrier L1 & Sum K1 = Sum L1 & K1 | the carrier of W1 = L1 ) defpred S1[ set , set ] means ( ( $1 in W1 & $2 = L1 . $1 ) or ( not $1 in W1 & $2 = 0. K ) ); reconsider L9 = L1 as Function of W1,K ; A1: for x being set st x in the carrier of V holds ex y being set st ( y in the carrier of K & S1[x,y] ) proof let x be set ; ::_thesis: ( x in the carrier of V implies ex y being set st ( y in the carrier of K & S1[x,y] ) ) assume x in the carrier of V ; ::_thesis: ex y being set st ( y in the carrier of K & S1[x,y] ) percases ( x in W1 or not x in W1 ) ; supposeA2: x in W1 ; ::_thesis: ex y being set st ( y in the carrier of K & S1[x,y] ) then reconsider x = x as Vector of W1 by STRUCT_0:def_5; S1[x,L1 . x] by A2; hence ex y being set st ( y in the carrier of K & S1[x,y] ) ; ::_thesis: verum end; suppose not x in W1 ; ::_thesis: ex y being set st ( y in the carrier of K & S1[x,y] ) hence ex y being set st ( y in the carrier of K & S1[x,y] ) ; ::_thesis: verum end; end; end; consider K1 being Function of V,K such that A3: for x being set st x in the carrier of V holds S1[x,K1 . x] from FUNCT_2:sch_1(A1); A4: the carrier of W1 c= the carrier of V by VECTSP_4:def_2; then reconsider C = Carrier L1 as finite Subset of V by XBOOLE_1:1; A5: now__::_thesis:_for_v_being_Vector_of_V_st_not_v_in_C_holds_ K1_._v_=_0._K let v be Vector of V; ::_thesis: ( not v in C implies K1 . v = 0. K ) assume A6: not v in C ; ::_thesis: K1 . v = 0. K ( ( S1[v,L1 . v] & v in the carrier of W1 ) or S1[v, 0. K] ) by STRUCT_0:def_5; then ( ( S1[v,L1 . v] & L1 . v = 0. K ) or S1[v, 0. K] ) by A6; hence K1 . v = 0. K by A3; ::_thesis: verum end; K1 is Element of Funcs ( the carrier of V, the carrier of K) by FUNCT_2:8; then reconsider K1 = K1 as Linear_Combination of V by A5, VECTSP_6:def_1; reconsider K9 = K1 | the carrier of W1 as Function of the carrier of W1, the carrier of K by A4, FUNCT_2:32; take K1 ; ::_thesis: ( Carrier K1 = Carrier L1 & Sum K1 = Sum L1 & K1 | the carrier of W1 = L1 ) now__::_thesis:_for_x_being_set_st_x_in_Carrier_K1_holds_ x_in_the_carrier_of_W1 let x be set ; ::_thesis: ( x in Carrier K1 implies x in the carrier of W1 ) assume that A7: x in Carrier K1 and A8: not x in the carrier of W1 ; ::_thesis: contradiction consider v being Vector of V such that A9: x = v and A10: K1 . v <> 0. K by A7; S1[v, 0. K] by A8, A9, STRUCT_0:def_5; hence contradiction by A3, A10; ::_thesis: verum end; then A11: Carrier K1 c= the carrier of W1 by TARSKI:def_3; now__::_thesis:_for_x_being_set_st_x_in_the_carrier_of_W1_holds_ L9_._x_=_K9_._x let x be set ; ::_thesis: ( x in the carrier of W1 implies L9 . x = K9 . x ) assume A12: x in the carrier of W1 ; ::_thesis: L9 . x = K9 . x S1[x,K1 . x] by A3, A4, A12; hence L9 . x = K9 . x by A12, FUNCT_1:49, STRUCT_0:def_5; ::_thesis: verum end; then L9 = K9 by FUNCT_2:12; hence ( Carrier K1 = Carrier L1 & Sum K1 = Sum L1 & K1 | the carrier of W1 = L1 ) by A11, VECTSP_9:7; ::_thesis: verum end; theorem :: MATRLIN2:22 for m being Nat for K being Field for V1 being finite-dimensional VectSp of K for b1 being OrdBasis of V1 holds ( rng (b1 | m) is linearly-independent Subset of V1 & ( for A being Subset of V1 st A = rng (b1 | m) holds b1 | m is OrdBasis of Lin A ) ) proof let m be Nat; ::_thesis: for K being Field for V1 being finite-dimensional VectSp of K for b1 being OrdBasis of V1 holds ( rng (b1 | m) is linearly-independent Subset of V1 & ( for A being Subset of V1 st A = rng (b1 | m) holds b1 | m is OrdBasis of Lin A ) ) let K be Field; ::_thesis: for V1 being finite-dimensional VectSp of K for b1 being OrdBasis of V1 holds ( rng (b1 | m) is linearly-independent Subset of V1 & ( for A being Subset of V1 st A = rng (b1 | m) holds b1 | m is OrdBasis of Lin A ) ) let V1 be finite-dimensional VectSp of K; ::_thesis: for b1 being OrdBasis of V1 holds ( rng (b1 | m) is linearly-independent Subset of V1 & ( for A being Subset of V1 st A = rng (b1 | m) holds b1 | m is OrdBasis of Lin A ) ) let b1 be OrdBasis of V1; ::_thesis: ( rng (b1 | m) is linearly-independent Subset of V1 & ( for A being Subset of V1 st A = rng (b1 | m) holds b1 | m is OrdBasis of Lin A ) ) reconsider RNG = rng b1 as Basis of V1 by MATRLIN:def_2; A1: RNG is linearly-independent by VECTSP_7:def_3; rng (b1 | m) c= RNG by RELAT_1:70; hence rng (b1 | m) is linearly-independent Subset of V1 by A1, VECTSP_7:1, XBOOLE_1:1; ::_thesis: for A being Subset of V1 st A = rng (b1 | m) holds b1 | m is OrdBasis of Lin A let A be Subset of V1; ::_thesis: ( A = rng (b1 | m) implies b1 | m is OrdBasis of Lin A ) assume A2: A = rng (b1 | m) ; ::_thesis: b1 | m is OrdBasis of Lin A A3: A c= the carrier of (Lin A) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in A or x in the carrier of (Lin A) ) assume x in A ; ::_thesis: x in the carrier of (Lin A) then x in Lin A by VECTSP_7:8; hence x in the carrier of (Lin A) by STRUCT_0:def_5; ::_thesis: verum end; A is linearly-independent by A1, A2, RELAT_1:70, VECTSP_7:1; then reconsider A9 = A as linearly-independent Subset of (Lin A) by A3, VECTSP_9:12; b1 is one-to-one by MATRLIN:def_2; then A4: b1 | m is one-to-one by FUNCT_1:52; Lin A9 = VectSpStr(# the carrier of (Lin A), the addF of (Lin A), the ZeroF of (Lin A), the lmult of (Lin A) #) by VECTSP_9:17; then ( rng (b1 | m) is Basis of Lin A & b1 | m is FinSequence of (Lin A) ) by A2, FINSEQ_1:def_4, VECTSP_7:def_3; hence b1 | m is OrdBasis of Lin A by A4, MATRLIN:def_2; ::_thesis: verum end; theorem :: MATRLIN2:23 for m being Nat for K being Field for V1 being finite-dimensional VectSp of K for b1 being OrdBasis of V1 holds ( rng (b1 /^ m) is linearly-independent Subset of V1 & ( for A being Subset of V1 st A = rng (b1 /^ m) holds b1 /^ m is OrdBasis of Lin A ) ) proof let m be Nat; ::_thesis: for K being Field for V1 being finite-dimensional VectSp of K for b1 being OrdBasis of V1 holds ( rng (b1 /^ m) is linearly-independent Subset of V1 & ( for A being Subset of V1 st A = rng (b1 /^ m) holds b1 /^ m is OrdBasis of Lin A ) ) let K be Field; ::_thesis: for V1 being finite-dimensional VectSp of K for b1 being OrdBasis of V1 holds ( rng (b1 /^ m) is linearly-independent Subset of V1 & ( for A being Subset of V1 st A = rng (b1 /^ m) holds b1 /^ m is OrdBasis of Lin A ) ) let V1 be finite-dimensional VectSp of K; ::_thesis: for b1 being OrdBasis of V1 holds ( rng (b1 /^ m) is linearly-independent Subset of V1 & ( for A being Subset of V1 st A = rng (b1 /^ m) holds b1 /^ m is OrdBasis of Lin A ) ) let b1 be OrdBasis of V1; ::_thesis: ( rng (b1 /^ m) is linearly-independent Subset of V1 & ( for A being Subset of V1 st A = rng (b1 /^ m) holds b1 /^ m is OrdBasis of Lin A ) ) reconsider RNG = rng b1 as Basis of V1 by MATRLIN:def_2; A1: RNG is linearly-independent by VECTSP_7:def_3; rng (b1 /^ m) c= RNG by FINSEQ_5:33; hence rng (b1 /^ m) is linearly-independent Subset of V1 by A1, VECTSP_7:1, XBOOLE_1:1; ::_thesis: for A being Subset of V1 st A = rng (b1 /^ m) holds b1 /^ m is OrdBasis of Lin A let A be Subset of V1; ::_thesis: ( A = rng (b1 /^ m) implies b1 /^ m is OrdBasis of Lin A ) assume A2: A = rng (b1 /^ m) ; ::_thesis: b1 /^ m is OrdBasis of Lin A A3: A c= the carrier of (Lin A) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in A or x in the carrier of (Lin A) ) assume x in A ; ::_thesis: x in the carrier of (Lin A) then x in Lin A by VECTSP_7:8; hence x in the carrier of (Lin A) by STRUCT_0:def_5; ::_thesis: verum end; A is linearly-independent by A1, A2, FINSEQ_5:33, VECTSP_7:1; then reconsider A9 = A as linearly-independent Subset of (Lin A) by A3, VECTSP_9:12; ( b1 is one-to-one & b1 = (b1 | m) ^ (b1 /^ m) ) by MATRLIN:def_2, RFINSEQ:8; then A4: b1 /^ m is one-to-one by FINSEQ_3:91; Lin A9 = VectSpStr(# the carrier of (Lin A), the addF of (Lin A), the ZeroF of (Lin A), the lmult of (Lin A) #) by VECTSP_9:17; then ( rng (b1 /^ m) is Basis of Lin A & b1 /^ m is FinSequence of (Lin A) ) by A2, FINSEQ_1:def_4, VECTSP_7:def_3; hence b1 /^ m is OrdBasis of Lin A by A4, MATRLIN:def_2; ::_thesis: verum end; theorem Th24: :: MATRLIN2:24 for K being Field for V1 being finite-dimensional VectSp of K for W1, W2 being Subspace of V1 st W1 /\ W2 = (0). V1 holds for b1 being OrdBasis of W1 for b2 being OrdBasis of W2 for b being OrdBasis of W1 + W2 st b = b1 ^ b2 holds for v, v1, v2 being Vector of (W1 + W2) for w1 being Vector of W1 for w2 being Vector of W2 st v = v1 + v2 & v1 = w1 & v2 = w2 holds v |-- b = (w1 |-- b1) ^ (w2 |-- b2) proof let K be Field; ::_thesis: for V1 being finite-dimensional VectSp of K for W1, W2 being Subspace of V1 st W1 /\ W2 = (0). V1 holds for b1 being OrdBasis of W1 for b2 being OrdBasis of W2 for b being OrdBasis of W1 + W2 st b = b1 ^ b2 holds for v, v1, v2 being Vector of (W1 + W2) for w1 being Vector of W1 for w2 being Vector of W2 st v = v1 + v2 & v1 = w1 & v2 = w2 holds v |-- b = (w1 |-- b1) ^ (w2 |-- b2) let V1 be finite-dimensional VectSp of K; ::_thesis: for W1, W2 being Subspace of V1 st W1 /\ W2 = (0). V1 holds for b1 being OrdBasis of W1 for b2 being OrdBasis of W2 for b being OrdBasis of W1 + W2 st b = b1 ^ b2 holds for v, v1, v2 being Vector of (W1 + W2) for w1 being Vector of W1 for w2 being Vector of W2 st v = v1 + v2 & v1 = w1 & v2 = w2 holds v |-- b = (w1 |-- b1) ^ (w2 |-- b2) let W1, W2 be Subspace of V1; ::_thesis: ( W1 /\ W2 = (0). V1 implies for b1 being OrdBasis of W1 for b2 being OrdBasis of W2 for b being OrdBasis of W1 + W2 st b = b1 ^ b2 holds for v, v1, v2 being Vector of (W1 + W2) for w1 being Vector of W1 for w2 being Vector of W2 st v = v1 + v2 & v1 = w1 & v2 = w2 holds v |-- b = (w1 |-- b1) ^ (w2 |-- b2) ) assume A1: W1 /\ W2 = (0). V1 ; ::_thesis: for b1 being OrdBasis of W1 for b2 being OrdBasis of W2 for b being OrdBasis of W1 + W2 st b = b1 ^ b2 holds for v, v1, v2 being Vector of (W1 + W2) for w1 being Vector of W1 for w2 being Vector of W2 st v = v1 + v2 & v1 = w1 & v2 = w2 holds v |-- b = (w1 |-- b1) ^ (w2 |-- b2) [#] ((0). V1) = {(0. V1)} by VECTSP_4:def_3; then A2: card ([#] ((0). V1)) = 1 by CARD_1:30; A3: (dim W1) + (dim W2) = (dim (W1 + W2)) + (dim (W1 /\ W2)) by VECTSP_9:32 .= (dim (W1 + W2)) + 0 by A1, A2, RANKNULL:5 ; let b1 be OrdBasis of W1; ::_thesis: for b2 being OrdBasis of W2 for b being OrdBasis of W1 + W2 st b = b1 ^ b2 holds for v, v1, v2 being Vector of (W1 + W2) for w1 being Vector of W1 for w2 being Vector of W2 st v = v1 + v2 & v1 = w1 & v2 = w2 holds v |-- b = (w1 |-- b1) ^ (w2 |-- b2) let b2 be OrdBasis of W2; ::_thesis: for b being OrdBasis of W1 + W2 st b = b1 ^ b2 holds for v, v1, v2 being Vector of (W1 + W2) for w1 being Vector of W1 for w2 being Vector of W2 st v = v1 + v2 & v1 = w1 & v2 = w2 holds v |-- b = (w1 |-- b1) ^ (w2 |-- b2) let b be OrdBasis of W1 + W2; ::_thesis: ( b = b1 ^ b2 implies for v, v1, v2 being Vector of (W1 + W2) for w1 being Vector of W1 for w2 being Vector of W2 st v = v1 + v2 & v1 = w1 & v2 = w2 holds v |-- b = (w1 |-- b1) ^ (w2 |-- b2) ) assume A4: b = b1 ^ b2 ; ::_thesis: for v, v1, v2 being Vector of (W1 + W2) for w1 being Vector of W1 for w2 being Vector of W2 st v = v1 + v2 & v1 = w1 & v2 = w2 holds v |-- b = (w1 |-- b1) ^ (w2 |-- b2) reconsider R = rng b as Basis of W1 + W2 by MATRLIN:def_2; let v, v1, v2 be Vector of (W1 + W2); ::_thesis: for w1 being Vector of W1 for w2 being Vector of W2 st v = v1 + v2 & v1 = w1 & v2 = w2 holds v |-- b = (w1 |-- b1) ^ (w2 |-- b2) let w1 be Vector of W1; ::_thesis: for w2 being Vector of W2 st v = v1 + v2 & v1 = w1 & v2 = w2 holds v |-- b = (w1 |-- b1) ^ (w2 |-- b2) let w2 be Vector of W2; ::_thesis: ( v = v1 + v2 & v1 = w1 & v2 = w2 implies v |-- b = (w1 |-- b1) ^ (w2 |-- b2) ) assume A5: ( v = v1 + v2 & v1 = w1 & v2 = w2 ) ; ::_thesis: v |-- b = (w1 |-- b1) ^ (w2 |-- b2) set wb2 = w2 |-- b2; consider L2 being Linear_Combination of W2 such that A6: w2 = Sum L2 and A7: Carrier L2 c= rng b2 and A8: for k being Nat st 1 <= k & k <= len (w2 |-- b2) holds (w2 |-- b2) /. k = L2 . (b2 /. k) by MATRLIN:def_7; A9: W2 is Subspace of W1 + W2 by VECTSP_5:7; then consider K2 being Linear_Combination of W1 + W2 such that A10: Carrier K2 = Carrier L2 and A11: Sum K2 = Sum L2 and A12: K2 | the carrier of W2 = L2 by Lm4; rng b2 c= R by A4, FINSEQ_1:30; then A13: Carrier K2 c= R by A7, A10, XBOOLE_1:1; set wb1 = w1 |-- b1; set vb = v |-- b; consider L1 being Linear_Combination of W1 such that A14: w1 = Sum L1 and A15: Carrier L1 c= rng b1 and A16: for k being Nat st 1 <= k & k <= len (w1 |-- b1) holds (w1 |-- b1) /. k = L1 . (b1 /. k) by MATRLIN:def_7; consider L being Linear_Combination of W1 + W2 such that A17: ( v = Sum L & Carrier L c= rng b ) and A18: for k being Nat st 1 <= k & k <= len (v |-- b) holds (v |-- b) /. k = L . (b /. k) by MATRLIN:def_7; A19: len (v |-- b) = len b by MATRLIN:def_7; then A20: dom (v |-- b) = dom b by FINSEQ_3:29; A21: len (w2 |-- b2) = len b2 by MATRLIN:def_7; then A22: dom (w2 |-- b2) = dom b2 by FINSEQ_3:29; A23: R is linearly-independent by VECTSP_7:def_3; A24: W1 is Subspace of W1 + W2 by VECTSP_5:7; then consider K1 being Linear_Combination of W1 + W2 such that A25: Carrier K1 = Carrier L1 and A26: Sum K1 = Sum L1 and A27: K1 | the carrier of W1 = L1 by Lm4; A28: len (w1 |-- b1) = len b1 by MATRLIN:def_7; then A29: dom (w1 |-- b1) = dom b1 by FINSEQ_3:29; A30: len ((w1 |-- b1) ^ (w2 |-- b2)) = (len (w1 |-- b1)) + (len (w2 |-- b2)) by FINSEQ_1:22; A31: ( len b1 = dim W1 & len b2 = dim W2 ) by Th21; A32: len b = dim (W1 + W2) by Th21; then A33: dom b = dom ((w1 |-- b1) ^ (w2 |-- b2)) by A28, A21, A31, A30, A3, FINSEQ_3:29; rng b1 c= R by A4, FINSEQ_1:29; then A34: Carrier K1 c= R by A15, A25, XBOOLE_1:1; then A35: L = K1 + K2 by A5, A14, A26, A6, A11, A17, A13, A23, MATRLIN:6; now__::_thesis:_for_k_being_Nat_st_1_<=_k_&_k_<=_len_(v_|--_b)_holds_ (v_|--_b)_._k_=_((w1_|--_b1)_^_(w2_|--_b2))_._k let k be Nat; ::_thesis: ( 1 <= k & k <= len (v |-- b) implies (v |-- b) . k = ((w1 |-- b1) ^ (w2 |-- b2)) . k ) assume A36: ( 1 <= k & k <= len (v |-- b) ) ; ::_thesis: (v |-- b) . k = ((w1 |-- b1) ^ (w2 |-- b2)) . k A37: k in dom ((w1 |-- b1) ^ (w2 |-- b2)) by A28, A21, A19, A31, A32, A30, A3, A36, FINSEQ_3:25; now__::_thesis:_((w1_|--_b1)_^_(w2_|--_b2))_._k_=_(v_|--_b)_._k percases ( k in dom (w1 |-- b1) or ex n being Nat st ( n in dom (w2 |-- b2) & k = (len (w1 |-- b1)) + n ) ) by A37, FINSEQ_1:25; supposeA38: k in dom (w1 |-- b1) ; ::_thesis: ((w1 |-- b1) ^ (w2 |-- b2)) . k = (v |-- b) . k then ( 1 <= k & k <= len (w1 |-- b1) ) by FINSEQ_3:25; then A39: L1 . (b1 /. k) = (w1 |-- b1) /. k by A16 .= (w1 |-- b1) . k by A38, PARTFUN1:def_6 .= ((w1 |-- b1) ^ (w2 |-- b2)) . k by A38, FINSEQ_1:def_7 ; reconsider b1k = b1 /. k as Vector of (W1 + W2) by A24, VECTSP_4:10; A40: K1 . (b1 /. k) = L1 . (b1 /. k) by A27, FUNCT_1:49; not b1 /. k in Carrier K2 proof A41: b1 /. k in W1 by STRUCT_0:def_5; assume A42: b1 /. k in Carrier K2 ; ::_thesis: contradiction then b1 /. k in W2 by A10, STRUCT_0:def_5; then b1 /. k in W1 /\ W2 by A41, VECTSP_5:3; then b1 /. k in the carrier of ((0). V1) by A1, STRUCT_0:def_5; then b1 /. k in {(0. V1)} by VECTSP_4:def_3; then b1 /. k = 0. V1 by TARSKI:def_1 .= 0. (W1 + W2) by VECTSP_4:11 ; hence contradiction by A13, A23, A42, VECTSP_7:2; ::_thesis: verum end; then K2 . b1k = 0. K ; then A43: L . b1k = (K1 . b1k) + (0. K) by A35, VECTSP_6:22 .= ((w1 |-- b1) ^ (w2 |-- b2)) . k by A39, A40, RLVECT_1:def_4 ; b1k = b1 . k by A29, A38, PARTFUN1:def_6 .= b . k by A4, A29, A38, FINSEQ_1:def_7 .= b /. k by A33, A37, PARTFUN1:def_6 ; hence ((w1 |-- b1) ^ (w2 |-- b2)) . k = (v |-- b) /. k by A18, A36, A43 .= (v |-- b) . k by A33, A20, A37, PARTFUN1:def_6 ; ::_thesis: verum end; suppose ex n being Nat st ( n in dom (w2 |-- b2) & k = (len (w1 |-- b1)) + n ) ; ::_thesis: ((w1 |-- b1) ^ (w2 |-- b2)) . k = (v |-- b) . k then consider n being Nat such that A44: n in dom (w2 |-- b2) and A45: k = (len (w1 |-- b1)) + n ; ( 1 <= n & n <= len (w2 |-- b2) ) by A44, FINSEQ_3:25; then A46: L2 . (b2 /. n) = (w2 |-- b2) /. n by A8 .= (w2 |-- b2) . n by A44, PARTFUN1:def_6 .= ((w1 |-- b1) ^ (w2 |-- b2)) . k by A44, A45, FINSEQ_1:def_7 ; reconsider b2n = b2 /. n as Vector of (W1 + W2) by A9, VECTSP_4:10; A47: K2 . (b2 /. n) = L2 . (b2 /. n) by A12, FUNCT_1:49; not b2 /. n in Carrier K1 proof assume A48: b2 /. n in Carrier K1 ; ::_thesis: contradiction then ( b2 /. n in W2 & b2 /. n in W1 ) by A25, STRUCT_0:def_5; then b2 /. n in W1 /\ W2 by VECTSP_5:3; then b2 /. n in the carrier of ((0). V1) by A1, STRUCT_0:def_5; then b2 /. n in {(0. V1)} by VECTSP_4:def_3; then b2 /. n = 0. V1 by TARSKI:def_1 .= 0. (W1 + W2) by VECTSP_4:11 ; hence contradiction by A34, A23, A48, VECTSP_7:2; ::_thesis: verum end; then K1 . b2n = 0. K ; then A49: L . b2n = (0. K) + (K2 . b2n) by A35, VECTSP_6:22 .= ((w1 |-- b1) ^ (w2 |-- b2)) . k by A46, A47, RLVECT_1:def_4 ; b2n = b2 . n by A22, A44, PARTFUN1:def_6 .= b . k by A4, A28, A22, A44, A45, FINSEQ_1:def_7 .= b /. k by A33, A37, PARTFUN1:def_6 ; hence ((w1 |-- b1) ^ (w2 |-- b2)) . k = (v |-- b) /. k by A18, A36, A49 .= (v |-- b) . k by A33, A20, A37, PARTFUN1:def_6 ; ::_thesis: verum end; end; end; hence (v |-- b) . k = ((w1 |-- b1) ^ (w2 |-- b2)) . k ; ::_thesis: verum end; hence v |-- b = (w1 |-- b1) ^ (w2 |-- b2) by A28, A21, A19, A31, A30, A3, Th21, FINSEQ_1:14; ::_thesis: verum end; theorem Th25: :: MATRLIN2:25 for K being Field for V1 being finite-dimensional VectSp of K for b1 being OrdBasis of V1 for W1 being Subspace of V1 st W1 = (Omega). V1 holds for w being Vector of W1 for v being Vector of V1 for w1 being OrdBasis of W1 st v = w & b1 = w1 holds v |-- b1 = w |-- w1 proof let K be Field; ::_thesis: for V1 being finite-dimensional VectSp of K for b1 being OrdBasis of V1 for W1 being Subspace of V1 st W1 = (Omega). V1 holds for w being Vector of W1 for v being Vector of V1 for w1 being OrdBasis of W1 st v = w & b1 = w1 holds v |-- b1 = w |-- w1 let V1 be finite-dimensional VectSp of K; ::_thesis: for b1 being OrdBasis of V1 for W1 being Subspace of V1 st W1 = (Omega). V1 holds for w being Vector of W1 for v being Vector of V1 for w1 being OrdBasis of W1 st v = w & b1 = w1 holds v |-- b1 = w |-- w1 let b1 be OrdBasis of V1; ::_thesis: for W1 being Subspace of V1 st W1 = (Omega). V1 holds for w being Vector of W1 for v being Vector of V1 for w1 being OrdBasis of W1 st v = w & b1 = w1 holds v |-- b1 = w |-- w1 let W1 be Subspace of V1; ::_thesis: ( W1 = (Omega). V1 implies for w being Vector of W1 for v being Vector of V1 for w1 being OrdBasis of W1 st v = w & b1 = w1 holds v |-- b1 = w |-- w1 ) assume A1: W1 = (Omega). V1 ; ::_thesis: for w being Vector of W1 for v being Vector of V1 for w1 being OrdBasis of W1 st v = w & b1 = w1 holds v |-- b1 = w |-- w1 let w be Vector of W1; ::_thesis: for v being Vector of V1 for w1 being OrdBasis of W1 st v = w & b1 = w1 holds v |-- b1 = w |-- w1 let v be Vector of V1; ::_thesis: for w1 being OrdBasis of W1 st v = w & b1 = w1 holds v |-- b1 = w |-- w1 let w1 be OrdBasis of W1; ::_thesis: ( v = w & b1 = w1 implies v |-- b1 = w |-- w1 ) assume that A2: v = w and A3: b1 = w1 ; ::_thesis: v |-- b1 = w |-- w1 consider KL being Linear_Combination of W1 such that A4: ( w = Sum KL & Carrier KL c= rng w1 ) and A5: for k being Nat st 1 <= k & k <= len (w |-- w1) holds (w |-- w1) /. k = KL . (w1 /. k) by MATRLIN:def_7; consider K1 being Linear_Combination of V1 such that A6: ( Carrier K1 = Carrier KL & Sum K1 = Sum KL ) and A7: K1 | the carrier of W1 = KL by Lm4; A8: len w1 = len (w |-- w1) by MATRLIN:def_7; now__::_thesis:_for_k_being_Nat_st_1_<=_k_&_k_<=_len_(w_|--_w1)_holds_ (w_|--_w1)_/._k_=_K1_._(b1_/._k) let k be Nat; ::_thesis: ( 1 <= k & k <= len (w |-- w1) implies (w |-- w1) /. k = K1 . (b1 /. k) ) assume A9: ( 1 <= k & k <= len (w |-- w1) ) ; ::_thesis: (w |-- w1) /. k = K1 . (b1 /. k) A10: k in dom w1 by A8, A9, FINSEQ_3:25; dom K1 = the carrier of W1 by A1, FUNCT_2:def_1; then KL = K1 by A7, RELAT_1:69; hence (w |-- w1) /. k = K1 . (w1 /. k) by A5, A9 .= K1 . (w1 . k) by A10, PARTFUN1:def_6 .= K1 . (b1 /. k) by A3, A10, PARTFUN1:def_6 ; ::_thesis: verum end; hence v |-- b1 = w |-- w1 by A2, A3, A4, A6, A8, MATRLIN:def_7; ::_thesis: verum end; theorem Th26: :: MATRLIN2:26 for K being Field for V1 being finite-dimensional VectSp of K for W1, W2 being Subspace of V1 st W1 /\ W2 = (0). V1 holds for w1 being OrdBasis of W1 for w2 being OrdBasis of W2 holds w1 ^ w2 is OrdBasis of W1 + W2 proof let K be Field; ::_thesis: for V1 being finite-dimensional VectSp of K for W1, W2 being Subspace of V1 st W1 /\ W2 = (0). V1 holds for w1 being OrdBasis of W1 for w2 being OrdBasis of W2 holds w1 ^ w2 is OrdBasis of W1 + W2 let V1 be finite-dimensional VectSp of K; ::_thesis: for W1, W2 being Subspace of V1 st W1 /\ W2 = (0). V1 holds for w1 being OrdBasis of W1 for w2 being OrdBasis of W2 holds w1 ^ w2 is OrdBasis of W1 + W2 let W1, W2 be Subspace of V1; ::_thesis: ( W1 /\ W2 = (0). V1 implies for w1 being OrdBasis of W1 for w2 being OrdBasis of W2 holds w1 ^ w2 is OrdBasis of W1 + W2 ) assume A1: W1 /\ W2 = (0). V1 ; ::_thesis: for w1 being OrdBasis of W1 for w2 being OrdBasis of W2 holds w1 ^ w2 is OrdBasis of W1 + W2 let w1 be OrdBasis of W1; ::_thesis: for w2 being OrdBasis of W2 holds w1 ^ w2 is OrdBasis of W1 + W2 let w2 be OrdBasis of W2; ::_thesis: w1 ^ w2 is OrdBasis of W1 + W2 reconsider R1 = rng w1 as Basis of W1 by MATRLIN:def_2; reconsider R2 = rng w2 as Basis of W2 by MATRLIN:def_2; A2: R1 \/ R2 = rng (w1 ^ w2) by FINSEQ_1:31; A3: R1 misses R2 proof assume R1 meets R2 ; ::_thesis: contradiction then consider x being set such that A4: x in R1 and A5: x in R2 by XBOOLE_0:3; ( x in W1 & x in W2 ) by A4, A5, STRUCT_0:def_5; then x in W1 /\ W2 by VECTSP_5:3; then x in the carrier of ((0). V1) by A1, STRUCT_0:def_5; then x in {(0. V1)} by VECTSP_4:def_3; then x = 0. V1 by TARSKI:def_1 .= 0. W1 by VECTSP_4:11 ; then not R1 is linearly-independent by A4, VECTSP_7:2; hence contradiction by VECTSP_7:def_3; ::_thesis: verum end; A6: R1 \/ R2 is Basis of W1 + W2 by A1, Th3; then reconsider w12 = w1 ^ w2 as FinSequence of (W1 + W2) by A2, FINSEQ_1:def_4; ( w1 is one-to-one & w2 is one-to-one ) by MATRLIN:def_2; then w12 is one-to-one by A3, FINSEQ_3:91; hence w1 ^ w2 is OrdBasis of W1 + W2 by A6, A2, MATRLIN:def_2; ::_thesis: verum end; begin definition let K be Field; let V1, V2 be finite-dimensional VectSp of K; let f be Function of V1,V2; let B1 be FinSequence of V1; let b2 be OrdBasis of V2; :: original: AutMt redefine func AutMt (f,B1,b2) -> Matrix of len B1, len b2,K; coherence AutMt (f,B1,b2) is Matrix of len B1, len b2,K proof reconsider A9 = AutMt (f,B1,b2) as Matrix of len (AutMt (f,B1,b2)), width (AutMt (f,B1,b2)),K by MATRIX_2:7; A1: len A9 = len B1 by MATRLIN:def_8; percases ( len B1 = 0 or len B1 > 0 ) ; supposeA2: len B1 = 0 ; ::_thesis: AutMt (f,B1,b2) is Matrix of len B1, len b2,K then AutMt (f,B1,b2) = {} by A1; hence AutMt (f,B1,b2) is Matrix of len B1, len b2,K by A2, MATRIX_1:13; ::_thesis: verum end; supposeA3: len B1 > 0 ; ::_thesis: AutMt (f,B1,b2) is Matrix of len B1, len b2,K A4: dom B1 = dom A9 by A1, FINSEQ_3:29; A5: dom B1 = Seg (len B1) by FINSEQ_1:def_3; A6: len B1 in Seg (len B1) by A3, FINSEQ_1:3; then (f . (B1 /. (len B1))) |-- b2 = A9 /. (len B1) by A5, MATRLIN:def_8 .= A9 . (len B1) by A3, A5, A4, FINSEQ_1:3, PARTFUN1:def_6 .= Line (A9,(len B1)) by A1, A6, MATRIX_2:8 ; then width A9 = len ((f . (B1 /. (len B1))) |-- b2) by CARD_1:def_7 .= len b2 by MATRLIN:def_7 ; hence AutMt (f,B1,b2) is Matrix of len B1, len b2,K by MATRLIN:def_8; ::_thesis: verum end; end; end; end; definition let S be 1-sorted ; let R be Relation; funcR | S -> set equals :: MATRLIN2:def 1 R | the carrier of S; coherence R | the carrier of S is set ; end; :: deftheorem defines | MATRLIN2:def_1_:_ for S being 1-sorted for R being Relation holds R | S = R | the carrier of S; theorem :: MATRLIN2:27 for K being Field for V1, V2 being finite-dimensional VectSp of K for b1 being OrdBasis of V1 for b2 being OrdBasis of V2 for f being linear-transformation of V1,V2 for W1, W2 being Subspace of V1 for U1, U2 being Subspace of V2 st ( dim W1 = 0 implies dim U1 = 0 ) & ( dim W2 = 0 implies dim U2 = 0 ) & V2 is_the_direct_sum_of U1,U2 holds for fW1 being linear-transformation of W1,U1 for fW2 being linear-transformation of W2,U2 st fW1 = f | W1 & fW2 = f | W2 holds for w1 being OrdBasis of W1 for w2 being OrdBasis of W2 for u1 being OrdBasis of U1 for u2 being OrdBasis of U2 st w1 ^ w2 = b1 & u1 ^ u2 = b2 holds AutMt (f,b1,b2) = block_diagonal (<*(AutMt (fW1,w1,u1)),(AutMt (fW2,w2,u2))*>,(0. K)) proof let K be Field; ::_thesis: for V1, V2 being finite-dimensional VectSp of K for b1 being OrdBasis of V1 for b2 being OrdBasis of V2 for f being linear-transformation of V1,V2 for W1, W2 being Subspace of V1 for U1, U2 being Subspace of V2 st ( dim W1 = 0 implies dim U1 = 0 ) & ( dim W2 = 0 implies dim U2 = 0 ) & V2 is_the_direct_sum_of U1,U2 holds for fW1 being linear-transformation of W1,U1 for fW2 being linear-transformation of W2,U2 st fW1 = f | W1 & fW2 = f | W2 holds for w1 being OrdBasis of W1 for w2 being OrdBasis of W2 for u1 being OrdBasis of U1 for u2 being OrdBasis of U2 st w1 ^ w2 = b1 & u1 ^ u2 = b2 holds AutMt (f,b1,b2) = block_diagonal (<*(AutMt (fW1,w1,u1)),(AutMt (fW2,w2,u2))*>,(0. K)) let V1, V2 be finite-dimensional VectSp of K; ::_thesis: for b1 being OrdBasis of V1 for b2 being OrdBasis of V2 for f being linear-transformation of V1,V2 for W1, W2 being Subspace of V1 for U1, U2 being Subspace of V2 st ( dim W1 = 0 implies dim U1 = 0 ) & ( dim W2 = 0 implies dim U2 = 0 ) & V2 is_the_direct_sum_of U1,U2 holds for fW1 being linear-transformation of W1,U1 for fW2 being linear-transformation of W2,U2 st fW1 = f | W1 & fW2 = f | W2 holds for w1 being OrdBasis of W1 for w2 being OrdBasis of W2 for u1 being OrdBasis of U1 for u2 being OrdBasis of U2 st w1 ^ w2 = b1 & u1 ^ u2 = b2 holds AutMt (f,b1,b2) = block_diagonal (<*(AutMt (fW1,w1,u1)),(AutMt (fW2,w2,u2))*>,(0. K)) let b1 be OrdBasis of V1; ::_thesis: for b2 being OrdBasis of V2 for f being linear-transformation of V1,V2 for W1, W2 being Subspace of V1 for U1, U2 being Subspace of V2 st ( dim W1 = 0 implies dim U1 = 0 ) & ( dim W2 = 0 implies dim U2 = 0 ) & V2 is_the_direct_sum_of U1,U2 holds for fW1 being linear-transformation of W1,U1 for fW2 being linear-transformation of W2,U2 st fW1 = f | W1 & fW2 = f | W2 holds for w1 being OrdBasis of W1 for w2 being OrdBasis of W2 for u1 being OrdBasis of U1 for u2 being OrdBasis of U2 st w1 ^ w2 = b1 & u1 ^ u2 = b2 holds AutMt (f,b1,b2) = block_diagonal (<*(AutMt (fW1,w1,u1)),(AutMt (fW2,w2,u2))*>,(0. K)) let b2 be OrdBasis of V2; ::_thesis: for f being linear-transformation of V1,V2 for W1, W2 being Subspace of V1 for U1, U2 being Subspace of V2 st ( dim W1 = 0 implies dim U1 = 0 ) & ( dim W2 = 0 implies dim U2 = 0 ) & V2 is_the_direct_sum_of U1,U2 holds for fW1 being linear-transformation of W1,U1 for fW2 being linear-transformation of W2,U2 st fW1 = f | W1 & fW2 = f | W2 holds for w1 being OrdBasis of W1 for w2 being OrdBasis of W2 for u1 being OrdBasis of U1 for u2 being OrdBasis of U2 st w1 ^ w2 = b1 & u1 ^ u2 = b2 holds AutMt (f,b1,b2) = block_diagonal (<*(AutMt (fW1,w1,u1)),(AutMt (fW2,w2,u2))*>,(0. K)) let f be linear-transformation of V1,V2; ::_thesis: for W1, W2 being Subspace of V1 for U1, U2 being Subspace of V2 st ( dim W1 = 0 implies dim U1 = 0 ) & ( dim W2 = 0 implies dim U2 = 0 ) & V2 is_the_direct_sum_of U1,U2 holds for fW1 being linear-transformation of W1,U1 for fW2 being linear-transformation of W2,U2 st fW1 = f | W1 & fW2 = f | W2 holds for w1 being OrdBasis of W1 for w2 being OrdBasis of W2 for u1 being OrdBasis of U1 for u2 being OrdBasis of U2 st w1 ^ w2 = b1 & u1 ^ u2 = b2 holds AutMt (f,b1,b2) = block_diagonal (<*(AutMt (fW1,w1,u1)),(AutMt (fW2,w2,u2))*>,(0. K)) let W1, W2 be Subspace of V1; ::_thesis: for U1, U2 being Subspace of V2 st ( dim W1 = 0 implies dim U1 = 0 ) & ( dim W2 = 0 implies dim U2 = 0 ) & V2 is_the_direct_sum_of U1,U2 holds for fW1 being linear-transformation of W1,U1 for fW2 being linear-transformation of W2,U2 st fW1 = f | W1 & fW2 = f | W2 holds for w1 being OrdBasis of W1 for w2 being OrdBasis of W2 for u1 being OrdBasis of U1 for u2 being OrdBasis of U2 st w1 ^ w2 = b1 & u1 ^ u2 = b2 holds AutMt (f,b1,b2) = block_diagonal (<*(AutMt (fW1,w1,u1)),(AutMt (fW2,w2,u2))*>,(0. K)) let U1, U2 be Subspace of V2; ::_thesis: ( ( dim W1 = 0 implies dim U1 = 0 ) & ( dim W2 = 0 implies dim U2 = 0 ) & V2 is_the_direct_sum_of U1,U2 implies for fW1 being linear-transformation of W1,U1 for fW2 being linear-transformation of W2,U2 st fW1 = f | W1 & fW2 = f | W2 holds for w1 being OrdBasis of W1 for w2 being OrdBasis of W2 for u1 being OrdBasis of U1 for u2 being OrdBasis of U2 st w1 ^ w2 = b1 & u1 ^ u2 = b2 holds AutMt (f,b1,b2) = block_diagonal (<*(AutMt (fW1,w1,u1)),(AutMt (fW2,w2,u2))*>,(0. K)) ) assume that A1: ( dim W1 = 0 implies dim U1 = 0 ) and A2: ( dim W2 = 0 implies dim U2 = 0 ) and A3: V2 is_the_direct_sum_of U1,U2 ; ::_thesis: for fW1 being linear-transformation of W1,U1 for fW2 being linear-transformation of W2,U2 st fW1 = f | W1 & fW2 = f | W2 holds for w1 being OrdBasis of W1 for w2 being OrdBasis of W2 for u1 being OrdBasis of U1 for u2 being OrdBasis of U2 st w1 ^ w2 = b1 & u1 ^ u2 = b2 holds AutMt (f,b1,b2) = block_diagonal (<*(AutMt (fW1,w1,u1)),(AutMt (fW2,w2,u2))*>,(0. K)) A4: U1 /\ U2 = (0). V2 by A3, VECTSP_5:def_4; let fW1 be linear-transformation of W1,U1; ::_thesis: for fW2 being linear-transformation of W2,U2 st fW1 = f | W1 & fW2 = f | W2 holds for w1 being OrdBasis of W1 for w2 being OrdBasis of W2 for u1 being OrdBasis of U1 for u2 being OrdBasis of U2 st w1 ^ w2 = b1 & u1 ^ u2 = b2 holds AutMt (f,b1,b2) = block_diagonal (<*(AutMt (fW1,w1,u1)),(AutMt (fW2,w2,u2))*>,(0. K)) let fW2 be linear-transformation of W2,U2; ::_thesis: ( fW1 = f | W1 & fW2 = f | W2 implies for w1 being OrdBasis of W1 for w2 being OrdBasis of W2 for u1 being OrdBasis of U1 for u2 being OrdBasis of U2 st w1 ^ w2 = b1 & u1 ^ u2 = b2 holds AutMt (f,b1,b2) = block_diagonal (<*(AutMt (fW1,w1,u1)),(AutMt (fW2,w2,u2))*>,(0. K)) ) assume that A5: fW1 = f | W1 and A6: fW2 = f | W2 ; ::_thesis: for w1 being OrdBasis of W1 for w2 being OrdBasis of W2 for u1 being OrdBasis of U1 for u2 being OrdBasis of U2 st w1 ^ w2 = b1 & u1 ^ u2 = b2 holds AutMt (f,b1,b2) = block_diagonal (<*(AutMt (fW1,w1,u1)),(AutMt (fW2,w2,u2))*>,(0. K)) let w1 be OrdBasis of W1; ::_thesis: for w2 being OrdBasis of W2 for u1 being OrdBasis of U1 for u2 being OrdBasis of U2 st w1 ^ w2 = b1 & u1 ^ u2 = b2 holds AutMt (f,b1,b2) = block_diagonal (<*(AutMt (fW1,w1,u1)),(AutMt (fW2,w2,u2))*>,(0. K)) let w2 be OrdBasis of W2; ::_thesis: for u1 being OrdBasis of U1 for u2 being OrdBasis of U2 st w1 ^ w2 = b1 & u1 ^ u2 = b2 holds AutMt (f,b1,b2) = block_diagonal (<*(AutMt (fW1,w1,u1)),(AutMt (fW2,w2,u2))*>,(0. K)) let u1 be OrdBasis of U1; ::_thesis: for u2 being OrdBasis of U2 st w1 ^ w2 = b1 & u1 ^ u2 = b2 holds AutMt (f,b1,b2) = block_diagonal (<*(AutMt (fW1,w1,u1)),(AutMt (fW2,w2,u2))*>,(0. K)) let u2 be OrdBasis of U2; ::_thesis: ( w1 ^ w2 = b1 & u1 ^ u2 = b2 implies AutMt (f,b1,b2) = block_diagonal (<*(AutMt (fW1,w1,u1)),(AutMt (fW2,w2,u2))*>,(0. K)) ) assume that A7: w1 ^ w2 = b1 and A8: u1 ^ u2 = b2 ; ::_thesis: AutMt (f,b1,b2) = block_diagonal (<*(AutMt (fW1,w1,u1)),(AutMt (fW2,w2,u2))*>,(0. K)) A9: len b1 = (len w1) + (len w2) by A7, FINSEQ_1:22; A10: U1 + U2 = (Omega). V2 by A3, VECTSP_5:def_4; set A = AutMt (f,b1,b2); A11: len b1 = len (AutMt (f,b1,b2)) by MATRLIN:def_8; set A2 = AutMt (fW2,w2,u2); A12: ( len w2 = dim W2 & len u2 = dim U2 ) by Th21; then A13: len w2 = len (AutMt (fW2,w2,u2)) by A2, MATRIX13:1; set A1 = AutMt (fW1,w1,u1); A14: ( len w1 = dim W1 & len u1 = dim U1 ) by Th21; then A15: len w1 = len (AutMt (fW1,w1,u1)) by A1, MATRIX13:1; A16: len u2 = width (AutMt (fW2,w2,u2)) by A2, A12, MATRIX13:1; A17: len u1 = width (AutMt (fW1,w1,u1)) by A1, A14, MATRIX13:1; A18: now__::_thesis:_for_i_being_Nat_holds_ (_(_i_in_dom_(AutMt_(fW1,w1,u1))_implies_Line_((AutMt_(f,b1,b2)),i)_=_(Line_((AutMt_(fW1,w1,u1)),i))_^_((width_(AutMt_(fW2,w2,u2)))_|->_(0._K))_)_&_(_i_in_dom_(AutMt_(fW2,w2,u2))_implies_Line_((AutMt_(f,b1,b2)),(i_+_(len_(AutMt_(fW1,w1,u1)))))_=_((width_(AutMt_(fW1,w1,u1)))_|->_(0._K))_^_(Line_((AutMt_(fW2,w2,u2)),i))_)_) reconsider uu = u1 ^ u2 as OrdBasis of U1 + U2 by A4, Th26; let i be Nat; ::_thesis: ( ( i in dom (AutMt (fW1,w1,u1)) implies Line ((AutMt (f,b1,b2)),i) = (Line ((AutMt (fW1,w1,u1)),i)) ^ ((width (AutMt (fW2,w2,u2))) |-> (0. K)) ) & ( i in dom (AutMt (fW2,w2,u2)) implies Line ((AutMt (f,b1,b2)),(i + (len (AutMt (fW1,w1,u1))))) = ((width (AutMt (fW1,w1,u1))) |-> (0. K)) ^ (Line ((AutMt (fW2,w2,u2)),i)) ) ) A19: dom (AutMt (f,b1,b2)) = Seg (len (AutMt (f,b1,b2))) by FINSEQ_1:def_3; reconsider fb = f . (b1 /. i), fbi = f . (b1 /. (i + (len (AutMt (fW1,w1,u1))))) as Vector of (U1 + U2) by A10; A20: dom (AutMt (f,b1,b2)) = dom b1 by A11, FINSEQ_3:29; A21: dom (AutMt (fW1,w1,u1)) = dom w1 by A15, FINSEQ_3:29; A22: dom fW1 = the carrier of W1 by FUNCT_2:def_1; thus ( i in dom (AutMt (fW1,w1,u1)) implies Line ((AutMt (f,b1,b2)),i) = (Line ((AutMt (fW1,w1,u1)),i)) ^ ((width (AutMt (fW2,w2,u2))) |-> (0. K)) ) ::_thesis: ( i in dom (AutMt (fW2,w2,u2)) implies Line ((AutMt (f,b1,b2)),(i + (len (AutMt (fW1,w1,u1))))) = ((width (AutMt (fW1,w1,u1))) |-> (0. K)) ^ (Line ((AutMt (fW2,w2,u2)),i)) ) proof assume A23: i in dom (AutMt (fW1,w1,u1)) ; ::_thesis: Line ((AutMt (f,b1,b2)),i) = (Line ((AutMt (fW1,w1,u1)),i)) ^ ((width (AutMt (fW2,w2,u2))) |-> (0. K)) A24: dom (AutMt (fW1,w1,u1)) = Seg (len (AutMt (fW1,w1,u1))) by FINSEQ_1:def_3; then A25: Line ((AutMt (fW1,w1,u1)),i) = (AutMt (fW1,w1,u1)) . i by A15, A23, MATRIX_2:8 .= (AutMt (fW1,w1,u1)) /. i by A23, PARTFUN1:def_6 .= (fW1 . (w1 /. i)) |-- u1 by A21, A23, MATRLIN:def_8 ; len (AutMt (fW1,w1,u1)) <= len (AutMt (f,b1,b2)) by A9, A15, A11, NAT_1:11; then A26: Seg (len (AutMt (fW1,w1,u1))) c= Seg (len (AutMt (f,b1,b2))) by FINSEQ_1:5; then b1 /. i = b1 . i by A19, A20, A23, A24, PARTFUN1:def_6 .= w1 . i by A7, A21, A23, FINSEQ_1:def_7 .= w1 /. i by A21, A23, PARTFUN1:def_6 ; then A27: fb = fW1 . (w1 /. i) by A5, A22, FUNCT_1:47; thus Line ((AutMt (f,b1,b2)),i) = (AutMt (f,b1,b2)) . i by A11, A23, A24, A26, MATRIX_2:8 .= (AutMt (f,b1,b2)) /. i by A19, A23, A24, A26, PARTFUN1:def_6 .= (f . (b1 /. i)) |-- b2 by A19, A20, A23, A24, A26, MATRLIN:def_8 .= fb |-- uu by A10, A8, Th25 .= (fb + (0. (U1 + U2))) |-- uu by RLVECT_1:def_4 .= ((fW1 . (w1 /. i)) |-- u1) ^ ((0. U2) |-- u2) by A4, A27, Th24, VECTSP_4:12 .= (Line ((AutMt (fW1,w1,u1)),i)) ^ ((width (AutMt (fW2,w2,u2))) |-> (0. K)) by A16, A25, Th20 ; ::_thesis: verum end; A28: dom (AutMt (fW2,w2,u2)) = dom w2 by A13, FINSEQ_3:29; A29: dom fW2 = the carrier of W2 by FUNCT_2:def_1; thus ( i in dom (AutMt (fW2,w2,u2)) implies Line ((AutMt (f,b1,b2)),(i + (len (AutMt (fW1,w1,u1))))) = ((width (AutMt (fW1,w1,u1))) |-> (0. K)) ^ (Line ((AutMt (fW2,w2,u2)),i)) ) ::_thesis: verum proof assume A30: i in dom (AutMt (fW2,w2,u2)) ; ::_thesis: Line ((AutMt (f,b1,b2)),(i + (len (AutMt (fW1,w1,u1))))) = ((width (AutMt (fW1,w1,u1))) |-> (0. K)) ^ (Line ((AutMt (fW2,w2,u2)),i)) A31: dom (AutMt (fW2,w2,u2)) = Seg (len (AutMt (fW2,w2,u2))) by FINSEQ_1:def_3; then A32: i + (len (AutMt (fW1,w1,u1))) in dom (AutMt (f,b1,b2)) by A9, A15, A13, A11, A19, A30, FINSEQ_1:60; b1 /. (i + (len (AutMt (fW1,w1,u1)))) = b1 . (i + (len (AutMt (fW1,w1,u1)))) by A9, A15, A13, A11, A19, A20, A30, A31, FINSEQ_1:60, PARTFUN1:def_6 .= w2 . i by A7, A15, A28, A30, FINSEQ_1:def_7 .= w2 /. i by A28, A30, PARTFUN1:def_6 ; then A33: fbi = fW2 . (w2 /. i) by A6, A29, FUNCT_1:47; A34: Line ((AutMt (fW2,w2,u2)),i) = (AutMt (fW2,w2,u2)) . i by A13, A30, A31, MATRIX_2:8 .= (AutMt (fW2,w2,u2)) /. i by A30, PARTFUN1:def_6 .= (fW2 . (w2 /. i)) |-- u2 by A28, A30, MATRLIN:def_8 ; thus Line ((AutMt (f,b1,b2)),(i + (len (AutMt (fW1,w1,u1))))) = (AutMt (f,b1,b2)) . (i + (len (AutMt (fW1,w1,u1)))) by A9, A15, A13, A11, A19, A30, A31, FINSEQ_1:60, MATRIX_2:8 .= (AutMt (f,b1,b2)) /. (i + (len (AutMt (fW1,w1,u1)))) by A9, A15, A13, A11, A19, A30, A31, FINSEQ_1:60, PARTFUN1:def_6 .= (f . (b1 /. (i + (len (AutMt (fW1,w1,u1)))))) |-- b2 by A20, A32, MATRLIN:def_8 .= fbi |-- uu by A10, A8, Th25 .= ((0. (U1 + U2)) + fbi) |-- uu by RLVECT_1:def_4 .= ((0. U1) |-- u1) ^ ((fW2 . (w2 /. i)) |-- u2) by A4, A33, Th24, VECTSP_4:12 .= ((width (AutMt (fW1,w1,u1))) |-> (0. K)) ^ (Line ((AutMt (fW2,w2,u2)),i)) by A17, A34, Th20 ; ::_thesis: verum end; end; set A12 = <*(AutMt (fW1,w1,u1)),(AutMt (fW2,w2,u2))*>; A35: ( Sum (Len <*(AutMt (fW1,w1,u1)),(AutMt (fW2,w2,u2))*>) = (len (AutMt (fW1,w1,u1))) + (len (AutMt (fW2,w2,u2))) & Sum (Width <*(AutMt (fW1,w1,u1)),(AutMt (fW2,w2,u2))*>) = (width (AutMt (fW1,w1,u1))) + (width (AutMt (fW2,w2,u2))) ) by MATRIXJ1:16, MATRIXJ1:20; len b2 = (len u1) + (len u2) by A8, FINSEQ_1:22; hence AutMt (f,b1,b2) = block_diagonal (<*(AutMt (fW1,w1,u1)),(AutMt (fW2,w2,u2))*>,(0. K)) by A9, A15, A13, A17, A16, A35, A18, MATRIXJ1:23; ::_thesis: verum end; definition let K be Field; let V1, V2 be finite-dimensional VectSp of K; let f be Function of V1,V2; let B1 be FinSequence of V1; let b2 be OrdBasis of V2; assume A1: len B1 = len b2 ; func AutEqMt (f,B1,b2) -> Matrix of len B1, len B1,K equals :Def2: :: MATRLIN2:def 2 AutMt (f,B1,b2); coherence AutMt (f,B1,b2) is Matrix of len B1, len B1,K by A1; end; :: deftheorem Def2 defines AutEqMt MATRLIN2:def_2_:_ for K being Field for V1, V2 being finite-dimensional VectSp of K for f being Function of V1,V2 for B1 being FinSequence of V1 for b2 being OrdBasis of V2 st len B1 = len b2 holds AutEqMt (f,B1,b2) = AutMt (f,B1,b2); theorem Th28: :: MATRLIN2:28 for K being Field for V1 being finite-dimensional VectSp of K for b1 being OrdBasis of V1 holds AutMt ((id V1),b1,b1) = 1. (K,(len b1)) proof let K be Field; ::_thesis: for V1 being finite-dimensional VectSp of K for b1 being OrdBasis of V1 holds AutMt ((id V1),b1,b1) = 1. (K,(len b1)) let V1 be finite-dimensional VectSp of K; ::_thesis: for b1 being OrdBasis of V1 holds AutMt ((id V1),b1,b1) = 1. (K,(len b1)) let b1 be OrdBasis of V1; ::_thesis: AutMt ((id V1),b1,b1) = 1. (K,(len b1)) set A = AutMt ((id V1),b1,b1); set ONE = 1. (K,(len b1)); A1: len (AutMt ((id V1),b1,b1)) = len b1 by MATRIX_1:def_2; A2: now__::_thesis:_for_i_being_Nat_st_1_<=_i_&_i_<=_len_b1_holds_ (AutMt_((id_V1),b1,b1))_._i_=_(1._(K,(len_b1)))_._i let i be Nat; ::_thesis: ( 1 <= i & i <= len b1 implies (AutMt ((id V1),b1,b1)) . i = (1. (K,(len b1))) . i ) assume A3: ( 1 <= i & i <= len b1 ) ; ::_thesis: (AutMt ((id V1),b1,b1)) . i = (1. (K,(len b1))) . i A4: i in dom b1 by A3, FINSEQ_3:25; A5: i in Seg (len b1) by A3, FINSEQ_1:1; i in dom (AutMt ((id V1),b1,b1)) by A1, A3, FINSEQ_3:25; hence (AutMt ((id V1),b1,b1)) . i = (AutMt ((id V1),b1,b1)) /. i by PARTFUN1:def_6 .= ((id V1) . (b1 /. i)) |-- b1 by A4, MATRLIN:def_8 .= (b1 /. i) |-- b1 by FUNCT_1:17 .= Line ((1. (K,(len b1))),i) by A4, Th19 .= (1. (K,(len b1))) . i by A5, MATRIX_2:8 ; ::_thesis: verum end; len (1. (K,(len b1))) = len b1 by MATRIX_1:def_2; hence AutMt ((id V1),b1,b1) = 1. (K,(len b1)) by A1, A2, FINSEQ_1:14; ::_thesis: verum end; theorem :: MATRLIN2:29 for K being Field for V1 being finite-dimensional VectSp of K for b1, b19 being OrdBasis of V1 holds ( AutEqMt ((id V1),b1,b19) is invertible & AutEqMt ((id V1),b19,b1) = (AutEqMt ((id V1),b1,b19)) ~ ) proof let K be Field; ::_thesis: for V1 being finite-dimensional VectSp of K for b1, b19 being OrdBasis of V1 holds ( AutEqMt ((id V1),b1,b19) is invertible & AutEqMt ((id V1),b19,b1) = (AutEqMt ((id V1),b1,b19)) ~ ) let V1 be finite-dimensional VectSp of K; ::_thesis: for b1, b19 being OrdBasis of V1 holds ( AutEqMt ((id V1),b1,b19) is invertible & AutEqMt ((id V1),b19,b1) = (AutEqMt ((id V1),b1,b19)) ~ ) let b1, b19 be OrdBasis of V1; ::_thesis: ( AutEqMt ((id V1),b1,b19) is invertible & AutEqMt ((id V1),b19,b1) = (AutEqMt ((id V1),b1,b19)) ~ ) set A = AutEqMt ((id V1),b1,b19); A1: 1_ K <> 0. K ; A2: len b1 = dim V1 by Th21 .= len b19 by Th21 ; then reconsider A9 = AutEqMt ((id V1),b19,b1) as Matrix of len b1, len b1,K ; A3: ( AutEqMt ((id V1),b1,b19) = AutMt ((id V1),b1,b19) & A9 = AutMt ((id V1),b19,b1) ) by A2, Def2; percases ( len b1 = 0 or (len b1) + 0 > 0 ) ; suppose len b1 = 0 ; ::_thesis: ( AutEqMt ((id V1),b1,b19) is invertible & AutEqMt ((id V1),b19,b1) = (AutEqMt ((id V1),b1,b19)) ~ ) then ( Det (AutEqMt ((id V1),b1,b19)) = 1_ K & A9 = (AutEqMt ((id V1),b1,b19)) ~ ) by MATRIXR2:41, MATRIX_1:35; hence ( AutEqMt ((id V1),b1,b19) is invertible & AutEqMt ((id V1),b19,b1) = (AutEqMt ((id V1),b1,b19)) ~ ) by A1, LAPLACE:34; ::_thesis: verum end; supposeA4: (len b1) + 0 > 0 ; ::_thesis: ( AutEqMt ((id V1),b1,b19) is invertible & AutEqMt ((id V1),b19,b1) = (AutEqMt ((id V1),b1,b19)) ~ ) dom (id V1) = the carrier of V1 by RELAT_1:45; then A5: (id V1) * (id V1) = id V1 by RELAT_1:52; len b1 = dim V1 by Th21; then len b1 = len b19 by Th21; then A6: (AutEqMt ((id V1),b1,b19)) * A9 = AutMt (((id V1) * (id V1)),b1,b1) by A3, A4, MATRLIN:41 .= 1. (K,(len b1)) by A5, Th28 ; len b1 >= 1 by A4, NAT_1:19; then 1_ K = Det ((AutEqMt ((id V1),b1,b19)) * A9) by A6, MATRIX_7:16 .= (Det (AutEqMt ((id V1),b1,b19))) * (Det A9) by A4, MATRIX11:62 ; then Det (AutEqMt ((id V1),b1,b19)) <> 0. K by VECTSP_1:12; then A7: AutEqMt ((id V1),b1,b19) is invertible by LAPLACE:34; then (AutEqMt ((id V1),b1,b19)) ~ is_reverse_of AutEqMt ((id V1),b1,b19) by MATRIX_6:def_4; then (AutEqMt ((id V1),b1,b19)) * ((AutEqMt ((id V1),b1,b19)) ~) = 1. (K,(len b1)) by MATRIX_6:def_2; hence ( AutEqMt ((id V1),b1,b19) is invertible & AutEqMt ((id V1),b19,b1) = (AutEqMt ((id V1),b1,b19)) ~ ) by A6, A7, MATRIX_8:19; ::_thesis: verum end; end; end; theorem Th30: :: MATRLIN2:30 for j being Nat for K being Field for V1 being finite-dimensional VectSp of K for b1 being OrdBasis of V1 for B1 being FinSequence of V1 for p1, p2 being FinSequence of K st len p1 = len p2 & len p1 = len B1 & len p1 > 0 & j in dom b1 & ( for i being Nat st i in dom p2 holds p2 . i = ((B1 /. i) |-- b1) . j ) holds p1 "*" p2 = ((Sum (lmlt (p1,B1))) |-- b1) . j proof let j be Nat; ::_thesis: for K being Field for V1 being finite-dimensional VectSp of K for b1 being OrdBasis of V1 for B1 being FinSequence of V1 for p1, p2 being FinSequence of K st len p1 = len p2 & len p1 = len B1 & len p1 > 0 & j in dom b1 & ( for i being Nat st i in dom p2 holds p2 . i = ((B1 /. i) |-- b1) . j ) holds p1 "*" p2 = ((Sum (lmlt (p1,B1))) |-- b1) . j let K be Field; ::_thesis: for V1 being finite-dimensional VectSp of K for b1 being OrdBasis of V1 for B1 being FinSequence of V1 for p1, p2 being FinSequence of K st len p1 = len p2 & len p1 = len B1 & len p1 > 0 & j in dom b1 & ( for i being Nat st i in dom p2 holds p2 . i = ((B1 /. i) |-- b1) . j ) holds p1 "*" p2 = ((Sum (lmlt (p1,B1))) |-- b1) . j let V1 be finite-dimensional VectSp of K; ::_thesis: for b1 being OrdBasis of V1 for B1 being FinSequence of V1 for p1, p2 being FinSequence of K st len p1 = len p2 & len p1 = len B1 & len p1 > 0 & j in dom b1 & ( for i being Nat st i in dom p2 holds p2 . i = ((B1 /. i) |-- b1) . j ) holds p1 "*" p2 = ((Sum (lmlt (p1,B1))) |-- b1) . j let b1 be OrdBasis of V1; ::_thesis: for B1 being FinSequence of V1 for p1, p2 being FinSequence of K st len p1 = len p2 & len p1 = len B1 & len p1 > 0 & j in dom b1 & ( for i being Nat st i in dom p2 holds p2 . i = ((B1 /. i) |-- b1) . j ) holds p1 "*" p2 = ((Sum (lmlt (p1,B1))) |-- b1) . j let B1 be FinSequence of V1; ::_thesis: for p1, p2 being FinSequence of K st len p1 = len p2 & len p1 = len B1 & len p1 > 0 & j in dom b1 & ( for i being Nat st i in dom p2 holds p2 . i = ((B1 /. i) |-- b1) . j ) holds p1 "*" p2 = ((Sum (lmlt (p1,B1))) |-- b1) . j let p1, p2 be FinSequence of K; ::_thesis: ( len p1 = len p2 & len p1 = len B1 & len p1 > 0 & j in dom b1 & ( for i being Nat st i in dom p2 holds p2 . i = ((B1 /. i) |-- b1) . j ) implies p1 "*" p2 = ((Sum (lmlt (p1,B1))) |-- b1) . j ) assume that A1: len p1 = len p2 and A2: len p1 = len B1 and A3: len p1 > 0 and A4: j in dom b1 and A5: for i being Nat st i in dom p2 holds p2 . i = ((B1 /. i) |-- b1) . j ; ::_thesis: p1 "*" p2 = ((Sum (lmlt (p1,B1))) |-- b1) . j deffunc H1( Nat, Nat) -> Element of the carrier of K = ((B1 /. $1) |-- b1) /. $2; consider M being Matrix of len p1, len b1,K such that A6: for i, j being Nat st [i,j] in Indices M holds M * (i,j) = H1(i,j) from MATRIX_1:sch_1(); A7: len M = len p1 by A3, MATRIX_1:23; then A8: dom p1 = dom M by FINSEQ_3:29; A9: width M = len b1 by A3, MATRIX_1:23; A10: dom b1 = Seg (len b1) by FINSEQ_1:def_3; A11: dom p1 = Seg (len p1) by FINSEQ_1:def_3; A12: dom p1 = dom p2 by A1, FINSEQ_3:29; A13: now__::_thesis:_for_i_being_Nat_st_1_<=_i_&_i_<=_len_p2_holds_ (Col_(M,j))_._i_=_p2_._i let i be Nat; ::_thesis: ( 1 <= i & i <= len p2 implies (Col (M,j)) . i = p2 . i ) assume ( 1 <= i & i <= len p2 ) ; ::_thesis: (Col (M,j)) . i = p2 . i then A14: i in dom p1 by A1, A11, FINSEQ_1:1; then A15: [i,j] in Indices M by A4, A9, A8, A10, ZFMISC_1:87; len ((B1 /. i) |-- b1) = len b1 by MATRLIN:def_7; then A16: dom ((B1 /. i) |-- b1) = dom b1 by FINSEQ_3:29; thus (Col (M,j)) . i = M * (i,j) by A8, A14, MATRIX_1:def_8 .= ((B1 /. i) |-- b1) /. j by A6, A15 .= ((B1 /. i) |-- b1) . j by A4, A16, PARTFUN1:def_6 .= p2 . i by A5, A12, A14 ; ::_thesis: verum end; deffunc H2( Nat) -> Element of the carrier of K = Sum (mlt (p1,(Col (M,$1)))); consider MC being FinSequence of K such that A17: ( len MC = len b1 & ( for j being Nat st j in dom MC holds MC . j = H2(j) ) ) from FINSEQ_2:sch_1(); A18: for j being Nat st j in dom MC holds MC /. j = H2(j) proof let j be Nat; ::_thesis: ( j in dom MC implies MC /. j = H2(j) ) assume A19: j in dom MC ; ::_thesis: MC /. j = H2(j) then MC . j = H2(j) by A17; hence MC /. j = H2(j) by A19, PARTFUN1:def_6; ::_thesis: verum end; A20: dom b1 = dom MC by A17, FINSEQ_3:29; A21: dom p1 = dom B1 by A2, FINSEQ_3:29; A22: now__::_thesis:_for_i_being_Nat_st_i_in_dom_B1_holds_ B1_/._i_=_Sum_(lmlt_((Line_(M,i)),b1)) let i be Nat; ::_thesis: ( i in dom B1 implies B1 /. i = Sum (lmlt ((Line (M,i)),b1)) ) assume A23: i in dom B1 ; ::_thesis: B1 /. i = Sum (lmlt ((Line (M,i)),b1)) A24: len (Line (M,i)) = width M by MATRIX_1:def_7; len ((B1 /. i) |-- b1) = len b1 by MATRLIN:def_7; then A25: dom (Line (M,i)) = dom ((B1 /. i) |-- b1) by A9, A24, FINSEQ_3:29; A26: dom (Line (M,i)) = Seg (width M) by A24, FINSEQ_1:def_3; A27: now__::_thesis:_for_k_being_Nat_st_k_in_dom_((B1_/._i)_|--_b1)_holds_ (Line_(M,i))_._k_=_((B1_/._i)_|--_b1)_._k let k be Nat; ::_thesis: ( k in dom ((B1 /. i) |-- b1) implies (Line (M,i)) . k = ((B1 /. i) |-- b1) . k ) assume A28: k in dom ((B1 /. i) |-- b1) ; ::_thesis: (Line (M,i)) . k = ((B1 /. i) |-- b1) . k A29: [i,k] in Indices M by A21, A8, A23, A25, A26, A28, ZFMISC_1:87; thus (Line (M,i)) . k = M * (i,k) by A25, A26, A28, MATRIX_1:def_7 .= ((B1 /. i) |-- b1) /. k by A6, A29 .= ((B1 /. i) |-- b1) . k by A28, PARTFUN1:def_6 ; ::_thesis: verum end; thus B1 /. i = Sum (lmlt (((B1 /. i) |-- b1),b1)) by MATRLIN:35 .= Sum (lmlt ((Line (M,i)),b1)) by A25, A27, FINSEQ_1:13 ; ::_thesis: verum end; A30: b1 <> {} by A4; A31: len (Col (M,j)) = len M by CARD_1:def_7; len ((Sum (lmlt (p1,B1))) |-- b1) = len b1 by MATRLIN:def_7; then dom ((Sum (lmlt (p1,B1))) |-- b1) = dom b1 by FINSEQ_3:29; hence ((Sum (lmlt (p1,B1))) |-- b1) . j = ((Sum (lmlt (p1,B1))) |-- b1) /. j by A4, PARTFUN1:def_6 .= ((Sum (lmlt (MC,b1))) |-- b1) /. j by A2, A3, A17, A18, A30, A22, MATRLIN:33 .= MC /. j by A17, MATRLIN:36 .= MC . j by A4, A20, PARTFUN1:def_6 .= Sum (mlt (p1,(Col (M,j)))) by A4, A17, A20 .= p1 "*" p2 by A1, A7, A31, A13, FINSEQ_1:14 ; ::_thesis: verum end; theorem Th31: :: MATRLIN2:31 for K being Field for V2, V1 being finite-dimensional VectSp of K for f being Function of V1,V2 for b1 being OrdBasis of V1 for b2 being OrdBasis of V2 for v1 being Element of V1 st len b1 > 0 & f is additive & f is homogeneous holds (LineVec2Mx (v1 |-- b1)) * (AutMt (f,b1,b2)) = LineVec2Mx ((f . v1) |-- b2) proof let K be Field; ::_thesis: for V2, V1 being finite-dimensional VectSp of K for f being Function of V1,V2 for b1 being OrdBasis of V1 for b2 being OrdBasis of V2 for v1 being Element of V1 st len b1 > 0 & f is additive & f is homogeneous holds (LineVec2Mx (v1 |-- b1)) * (AutMt (f,b1,b2)) = LineVec2Mx ((f . v1) |-- b2) let V2, V1 be finite-dimensional VectSp of K; ::_thesis: for f being Function of V1,V2 for b1 being OrdBasis of V1 for b2 being OrdBasis of V2 for v1 being Element of V1 st len b1 > 0 & f is additive & f is homogeneous holds (LineVec2Mx (v1 |-- b1)) * (AutMt (f,b1,b2)) = LineVec2Mx ((f . v1) |-- b2) let f be Function of V1,V2; ::_thesis: for b1 being OrdBasis of V1 for b2 being OrdBasis of V2 for v1 being Element of V1 st len b1 > 0 & f is additive & f is homogeneous holds (LineVec2Mx (v1 |-- b1)) * (AutMt (f,b1,b2)) = LineVec2Mx ((f . v1) |-- b2) let b1 be OrdBasis of V1; ::_thesis: for b2 being OrdBasis of V2 for v1 being Element of V1 st len b1 > 0 & f is additive & f is homogeneous holds (LineVec2Mx (v1 |-- b1)) * (AutMt (f,b1,b2)) = LineVec2Mx ((f . v1) |-- b2) let b2 be OrdBasis of V2; ::_thesis: for v1 being Element of V1 st len b1 > 0 & f is additive & f is homogeneous holds (LineVec2Mx (v1 |-- b1)) * (AutMt (f,b1,b2)) = LineVec2Mx ((f . v1) |-- b2) let v1 be Element of V1; ::_thesis: ( len b1 > 0 & f is additive & f is homogeneous implies (LineVec2Mx (v1 |-- b1)) * (AutMt (f,b1,b2)) = LineVec2Mx ((f . v1) |-- b2) ) assume that A1: len b1 > 0 and A2: ( f is additive & f is homogeneous ) ; ::_thesis: (LineVec2Mx (v1 |-- b1)) * (AutMt (f,b1,b2)) = LineVec2Mx ((f . v1) |-- b2) set A = AutMt (f,b1,b2); set fb = (f . v1) |-- b2; set vb = v1 |-- b1; set L = LineVec2Mx (v1 |-- b1); set LA = (LineVec2Mx (v1 |-- b1)) * (AutMt (f,b1,b2)); set Lf = LineVec2Mx ((f . v1) |-- b2); A3: len (AutMt (f,b1,b2)) = len b1 by MATRLIN:def_8; len ((f . v1) |-- b2) = len b2 by MATRLIN:def_7; then A4: width (LineVec2Mx ((f . v1) |-- b2)) = len b2 by MATRIX_1:23; A5: len (v1 |-- b1) = len b1 by MATRLIN:def_7; then A6: width (LineVec2Mx (v1 |-- b1)) = len b1 by MATRIX_1:23; len (LineVec2Mx (v1 |-- b1)) = 1 by MATRIX_1:23; then A7: len ((LineVec2Mx (v1 |-- b1)) * (AutMt (f,b1,b2))) = 1 by A6, A3, MATRIX_3:def_4; A8: width (AutMt (f,b1,b2)) = len b2 by A1, MATRLIN:39; then A9: width ((LineVec2Mx (v1 |-- b1)) * (AutMt (f,b1,b2))) = len b2 by A6, A3, MATRIX_3:def_4; A10: now__::_thesis:_for_i,_j_being_Nat_st_[i,j]_in_Indices_((LineVec2Mx_(v1_|--_b1))_*_(AutMt_(f,b1,b2)))_holds_ (LineVec2Mx_((f_._v1)_|--_b2))_*_(i,j)_=_((LineVec2Mx_(v1_|--_b1))_*_(AutMt_(f,b1,b2)))_*_(i,j) A11: dom b2 = Seg (len b2) by FINSEQ_1:def_3; A12: dom ((LineVec2Mx (v1 |-- b1)) * (AutMt (f,b1,b2))) = Seg 1 by A7, FINSEQ_1:def_3; A13: len (f * b1) = len b1 by FINSEQ_2:33; let i, j be Nat; ::_thesis: ( [i,j] in Indices ((LineVec2Mx (v1 |-- b1)) * (AutMt (f,b1,b2))) implies (LineVec2Mx ((f . v1) |-- b2)) * (i,j) = ((LineVec2Mx (v1 |-- b1)) * (AutMt (f,b1,b2))) * (i,j) ) assume A14: [i,j] in Indices ((LineVec2Mx (v1 |-- b1)) * (AutMt (f,b1,b2))) ; ::_thesis: (LineVec2Mx ((f . v1) |-- b2)) * (i,j) = ((LineVec2Mx (v1 |-- b1)) * (AutMt (f,b1,b2))) * (i,j) A15: j in Seg (len b2) by A9, A14, ZFMISC_1:87; i in dom ((LineVec2Mx (v1 |-- b1)) * (AutMt (f,b1,b2))) by A14, ZFMISC_1:87; then A16: i = 1 by A12, FINSEQ_1:2, TARSKI:def_1; A17: len (Col ((AutMt (f,b1,b2)),j)) = len (AutMt (f,b1,b2)) by CARD_1:def_7; A18: now__::_thesis:_for_k_being_Nat_st_k_in_dom_(Col_((AutMt_(f,b1,b2)),j))_holds_ (Col_((AutMt_(f,b1,b2)),j))_._k_=_(((f_*_b1)_/._k)_|--_b2)_._j A19: dom (f * b1) = dom b1 by A13, FINSEQ_3:29; A20: dom (AutMt (f,b1,b2)) = dom (Col ((AutMt (f,b1,b2)),j)) by A17, FINSEQ_3:29; let k be Nat; ::_thesis: ( k in dom (Col ((AutMt (f,b1,b2)),j)) implies (Col ((AutMt (f,b1,b2)),j)) . k = (((f * b1) /. k) |-- b2) . j ) assume A21: k in dom (Col ((AutMt (f,b1,b2)),j)) ; ::_thesis: (Col ((AutMt (f,b1,b2)),j)) . k = (((f * b1) /. k) |-- b2) . j A22: ( dom (AutMt (f,b1,b2)) = Seg (len (AutMt (f,b1,b2))) & (AutMt (f,b1,b2)) . k = (AutMt (f,b1,b2)) /. k ) by A21, A20, FINSEQ_1:def_3, PARTFUN1:def_6; A23: dom (AutMt (f,b1,b2)) = dom b1 by A3, FINSEQ_3:29; then A24: f . (b1 /. k) = f . (b1 . k) by A21, A20, PARTFUN1:def_6 .= (f * b1) . k by A21, A20, A23, FUNCT_1:13 .= (f * b1) /. k by A21, A20, A23, A19, PARTFUN1:def_6 ; thus (Col ((AutMt (f,b1,b2)),j)) . k = (AutMt (f,b1,b2)) * (k,j) by A21, A20, MATRIX_1:def_8 .= (Line ((AutMt (f,b1,b2)),k)) . j by A8, A15, MATRIX_1:def_7 .= ((AutMt (f,b1,b2)) /. k) . j by A3, A21, A20, A22, MATRIX_2:8 .= (((f * b1) /. k) |-- b2) . j by A21, A20, A23, A24, MATRLIN:def_8 ; ::_thesis: verum end; thus (LineVec2Mx ((f . v1) |-- b2)) * (i,j) = (Line ((LineVec2Mx ((f . v1) |-- b2)),i)) . j by A4, A15, MATRIX_1:def_7 .= ((f . v1) |-- b2) . j by A16, MATRIX15:25 .= ((f . (Sum (lmlt ((v1 |-- b1),b1)))) |-- b2) . j by MATRLIN:35 .= ((Sum (lmlt ((v1 |-- b1),(f * b1)))) |-- b2) . j by A2, A5, MATRLIN:18 .= (v1 |-- b1) "*" (Col ((AutMt (f,b1,b2)),j)) by A1, A5, A3, A11, A15, A13, A17, A18, Th30 .= (Line ((LineVec2Mx (v1 |-- b1)),1)) "*" (Col ((AutMt (f,b1,b2)),j)) by MATRIX15:25 .= ((LineVec2Mx (v1 |-- b1)) * (AutMt (f,b1,b2))) * (i,j) by A6, A3, A14, A16, MATRIX_3:def_4 ; ::_thesis: verum end; len (LineVec2Mx ((f . v1) |-- b2)) = 1 by MATRIX_1:23; hence (LineVec2Mx (v1 |-- b1)) * (AutMt (f,b1,b2)) = LineVec2Mx ((f . v1) |-- b2) by A7, A9, A4, A10, MATRIX_1:21; ::_thesis: verum end; begin definition let K be Field; let V1, V2 be finite-dimensional VectSp of K; let b1 be OrdBasis of V1; let B2 be FinSequence of V2; let M be Matrix of len b1, len B2,K; func Mx2Tran (M,b1,B2) -> Function of V1,V2 means :Def3: :: MATRLIN2:def 3 for v being Vector of V1 holds it . v = Sum (lmlt ((Line (((LineVec2Mx (v |-- b1)) * M),1)),B2)); existence ex b1 being Function of V1,V2 st for v being Vector of V1 holds b1 . v = Sum (lmlt ((Line (((LineVec2Mx (v |-- b1)) * M),1)),B2)) proof deffunc H1( Element of V1) -> Element of the carrier of V2 = Sum (lmlt ((Line (((LineVec2Mx ($1 |-- b1)) * M),1)),B2)); consider f being Function of V1,V2 such that A1: for x being Element of V1 holds f . x = H1(x) from FUNCT_2:sch_4(); take f ; ::_thesis: for v being Vector of V1 holds f . v = Sum (lmlt ((Line (((LineVec2Mx (v |-- b1)) * M),1)),B2)) thus for v being Vector of V1 holds f . v = Sum (lmlt ((Line (((LineVec2Mx (v |-- b1)) * M),1)),B2)) by A1; ::_thesis: verum end; uniqueness for b1, b2 being Function of V1,V2 st ( for v being Vector of V1 holds b1 . v = Sum (lmlt ((Line (((LineVec2Mx (v |-- b1)) * M),1)),B2)) ) & ( for v being Vector of V1 holds b2 . v = Sum (lmlt ((Line (((LineVec2Mx (v |-- b1)) * M),1)),B2)) ) holds b1 = b2 proof let F1, F2 be Function of V1,V2; ::_thesis: ( ( for v being Vector of V1 holds F1 . v = Sum (lmlt ((Line (((LineVec2Mx (v |-- b1)) * M),1)),B2)) ) & ( for v being Vector of V1 holds F2 . v = Sum (lmlt ((Line (((LineVec2Mx (v |-- b1)) * M),1)),B2)) ) implies F1 = F2 ) assume that A2: for v being Vector of V1 holds F1 . v = Sum (lmlt ((Line (((LineVec2Mx (v |-- b1)) * M),1)),B2)) and A3: for v being Vector of V1 holds F2 . v = Sum (lmlt ((Line (((LineVec2Mx (v |-- b1)) * M),1)),B2)) ; ::_thesis: F1 = F2 now__::_thesis:_for_x_being_set_st_x_in_the_carrier_of_V1_holds_ F1_._x_=_F2_._x let x be set ; ::_thesis: ( x in the carrier of V1 implies F1 . x = F2 . x ) assume x in the carrier of V1 ; ::_thesis: F1 . x = F2 . x then reconsider v = x as Vector of V1 ; thus F1 . x = Sum (lmlt ((Line (((LineVec2Mx (v |-- b1)) * M),1)),B2)) by A2 .= F2 . x by A3 ; ::_thesis: verum end; hence F1 = F2 by FUNCT_2:12; ::_thesis: verum end; end; :: deftheorem Def3 defines Mx2Tran MATRLIN2:def_3_:_ for K being Field for V1, V2 being finite-dimensional VectSp of K for b1 being OrdBasis of V1 for B2 being FinSequence of V2 for M being Matrix of len b1, len B2,K for b7 being Function of V1,V2 holds ( b7 = Mx2Tran (M,b1,B2) iff for v being Vector of V1 holds b7 . v = Sum (lmlt ((Line (((LineVec2Mx (v |-- b1)) * M),1)),B2)) ); theorem Th32: :: MATRLIN2:32 for K being Field for V2, V1 being finite-dimensional VectSp of K for b1 being OrdBasis of V1 for b2 being OrdBasis of V2 for v1 being Element of V1 for M being Matrix of len b1, len b2,K st len b1 > 0 holds LineVec2Mx (((Mx2Tran (M,b1,b2)) . v1) |-- b2) = (LineVec2Mx (v1 |-- b1)) * M proof let K be Field; ::_thesis: for V2, V1 being finite-dimensional VectSp of K for b1 being OrdBasis of V1 for b2 being OrdBasis of V2 for v1 being Element of V1 for M being Matrix of len b1, len b2,K st len b1 > 0 holds LineVec2Mx (((Mx2Tran (M,b1,b2)) . v1) |-- b2) = (LineVec2Mx (v1 |-- b1)) * M let V2, V1 be finite-dimensional VectSp of K; ::_thesis: for b1 being OrdBasis of V1 for b2 being OrdBasis of V2 for v1 being Element of V1 for M being Matrix of len b1, len b2,K st len b1 > 0 holds LineVec2Mx (((Mx2Tran (M,b1,b2)) . v1) |-- b2) = (LineVec2Mx (v1 |-- b1)) * M let b1 be OrdBasis of V1; ::_thesis: for b2 being OrdBasis of V2 for v1 being Element of V1 for M being Matrix of len b1, len b2,K st len b1 > 0 holds LineVec2Mx (((Mx2Tran (M,b1,b2)) . v1) |-- b2) = (LineVec2Mx (v1 |-- b1)) * M let b2 be OrdBasis of V2; ::_thesis: for v1 being Element of V1 for M being Matrix of len b1, len b2,K st len b1 > 0 holds LineVec2Mx (((Mx2Tran (M,b1,b2)) . v1) |-- b2) = (LineVec2Mx (v1 |-- b1)) * M let v1 be Element of V1; ::_thesis: for M being Matrix of len b1, len b2,K st len b1 > 0 holds LineVec2Mx (((Mx2Tran (M,b1,b2)) . v1) |-- b2) = (LineVec2Mx (v1 |-- b1)) * M set L = LineVec2Mx (v1 |-- b1); A1: ( width (LineVec2Mx (v1 |-- b1)) = len (v1 |-- b1) & len (v1 |-- b1) = len b1 ) by MATRIX_1:23, MATRLIN:def_7; let M be Matrix of len b1, len b2,K; ::_thesis: ( len b1 > 0 implies LineVec2Mx (((Mx2Tran (M,b1,b2)) . v1) |-- b2) = (LineVec2Mx (v1 |-- b1)) * M ) assume A2: len b1 > 0 ; ::_thesis: LineVec2Mx (((Mx2Tran (M,b1,b2)) . v1) |-- b2) = (LineVec2Mx (v1 |-- b1)) * M A3: len M = len b1 by A2, MATRIX_1:23; set LM = (LineVec2Mx (v1 |-- b1)) * M; width M = len b2 by A2, MATRIX_1:23; then width ((LineVec2Mx (v1 |-- b1)) * M) = len b2 by A1, A3, MATRIX_3:def_4; then len (Line (((LineVec2Mx (v1 |-- b1)) * M),1)) = len b2 by CARD_1:def_7; then A4: (Sum (lmlt ((Line (((LineVec2Mx (v1 |-- b1)) * M),1)),b2))) |-- b2 = Line (((LineVec2Mx (v1 |-- b1)) * M),1) by MATRLIN:36; len (LineVec2Mx (v1 |-- b1)) = 1 by MATRIX_1:23; then len ((LineVec2Mx (v1 |-- b1)) * M) = 1 by A1, A3, MATRIX_3:def_4; hence (LineVec2Mx (v1 |-- b1)) * M = LineVec2Mx ((Sum (lmlt ((Line (((LineVec2Mx (v1 |-- b1)) * M),1)),b2))) |-- b2) by A4, MATRIX15:25 .= LineVec2Mx (((Mx2Tran (M,b1,b2)) . v1) |-- b2) by Def3 ; ::_thesis: verum end; theorem Th33: :: MATRLIN2:33 for K being Field for V1, V2 being finite-dimensional VectSp of K for b1 being OrdBasis of V1 for B2 being FinSequence of V2 for v1 being Element of V1 for M being Matrix of len b1, len B2,K st len b1 = 0 holds (Mx2Tran (M,b1,B2)) . v1 = 0. V2 proof let K be Field; ::_thesis: for V1, V2 being finite-dimensional VectSp of K for b1 being OrdBasis of V1 for B2 being FinSequence of V2 for v1 being Element of V1 for M being Matrix of len b1, len B2,K st len b1 = 0 holds (Mx2Tran (M,b1,B2)) . v1 = 0. V2 let V1, V2 be finite-dimensional VectSp of K; ::_thesis: for b1 being OrdBasis of V1 for B2 being FinSequence of V2 for v1 being Element of V1 for M being Matrix of len b1, len B2,K st len b1 = 0 holds (Mx2Tran (M,b1,B2)) . v1 = 0. V2 let b1 be OrdBasis of V1; ::_thesis: for B2 being FinSequence of V2 for v1 being Element of V1 for M being Matrix of len b1, len B2,K st len b1 = 0 holds (Mx2Tran (M,b1,B2)) . v1 = 0. V2 let B2 be FinSequence of V2; ::_thesis: for v1 being Element of V1 for M being Matrix of len b1, len B2,K st len b1 = 0 holds (Mx2Tran (M,b1,B2)) . v1 = 0. V2 let v1 be Element of V1; ::_thesis: for M being Matrix of len b1, len B2,K st len b1 = 0 holds (Mx2Tran (M,b1,B2)) . v1 = 0. V2 let M be Matrix of len b1, len B2,K; ::_thesis: ( len b1 = 0 implies (Mx2Tran (M,b1,B2)) . v1 = 0. V2 ) assume A1: len b1 = 0 ; ::_thesis: (Mx2Tran (M,b1,B2)) . v1 = 0. V2 set L = LineVec2Mx (v1 |-- b1); set LM = (LineVec2Mx (v1 |-- b1)) * M; set LL = Line (((LineVec2Mx (v1 |-- b1)) * M),1); A2: ( width (LineVec2Mx (v1 |-- b1)) = len (v1 |-- b1) & len (v1 |-- b1) = len b1 ) by MATRIX_1:23, MATRLIN:def_7; A3: len M = len b1 by MATRIX_1:def_2; then width M = 0 by A1, MATRIX_1:def_3; then width ((LineVec2Mx (v1 |-- b1)) * M) = 0 by A2, A3, MATRIX_3:def_4; then A4: dom (Line (((LineVec2Mx (v1 |-- b1)) * M),1)) = {} ; dom (lmlt ((Line (((LineVec2Mx (v1 |-- b1)) * M),1)),B2)) = (dom (Line (((LineVec2Mx (v1 |-- b1)) * M),1))) /\ (dom B2) by Lm1; then lmlt ((Line (((LineVec2Mx (v1 |-- b1)) * M),1)),B2) = <*> the carrier of V2 by A4; then Sum (lmlt ((Line (((LineVec2Mx (v1 |-- b1)) * M),1)),B2)) = 0. V2 by RLVECT_1:43; hence (Mx2Tran (M,b1,B2)) . v1 = 0. V2 by Def3; ::_thesis: verum end; Lm5: for K being Field for A, B being Matrix of K st width A = width B holds for i being Nat st 1 <= i & i <= len A holds Line ((A + B),i) = (Line (A,i)) + (Line (B,i)) proof let K be Field; ::_thesis: for A, B being Matrix of K st width A = width B holds for i being Nat st 1 <= i & i <= len A holds Line ((A + B),i) = (Line (A,i)) + (Line (B,i)) let A, B be Matrix of K; ::_thesis: ( width A = width B implies for i being Nat st 1 <= i & i <= len A holds Line ((A + B),i) = (Line (A,i)) + (Line (B,i)) ) assume A1: width A = width B ; ::_thesis: for i being Nat st 1 <= i & i <= len A holds Line ((A + B),i) = (Line (A,i)) + (Line (B,i)) let i be Nat; ::_thesis: ( 1 <= i & i <= len A implies Line ((A + B),i) = (Line (A,i)) + (Line (B,i)) ) assume ( 1 <= i & i <= len A ) ; ::_thesis: Line ((A + B),i) = (Line (A,i)) + (Line (B,i)) then A2: i in dom A by FINSEQ_3:25; reconsider LB = Line (B,i) as Element of (width A) -tuples_on the carrier of K by A1; percases ( width A > 0 or width A = 0 ) ; suppose width A > 0 ; ::_thesis: Line ((A + B),i) = (Line (A,i)) + (Line (B,i)) then width A in Seg (width A) by FINSEQ_1:3; then [i,(width A)] in Indices A by A2, ZFMISC_1:87; hence Line ((A + B),i) = (Line (A,i)) + (Line (B,i)) by A1, MATRIX_4:59; ::_thesis: verum end; supposeA3: width A = 0 ; ::_thesis: Line ((A + B),i) = (Line (A,i)) + (Line (B,i)) then len ((Line (A,i)) + LB) = 0 ; then A4: (Line (A,i)) + (Line (B,i)) = {} ; width (A + B) = 0 by A3, MATRIX_3:def_3; hence Line ((A + B),i) = (Line (A,i)) + (Line (B,i)) by A4; ::_thesis: verum end; end; end; registration let K be Field; let V1, V2 be finite-dimensional VectSp of K; let b1 be OrdBasis of V1; let B2 be FinSequence of V2; let M be Matrix of len b1, len B2,K; cluster Mx2Tran (M,b1,B2) -> additive homogeneous ; coherence ( Mx2Tran (M,b1,B2) is homogeneous & Mx2Tran (M,b1,B2) is additive ) proof set Mx = Mx2Tran (M,b1,B2); percases ( len b1 = 0 or len b1 > 0 ) ; supposeA1: len b1 = 0 ; ::_thesis: ( Mx2Tran (M,b1,B2) is homogeneous & Mx2Tran (M,b1,B2) is additive ) A2: now__::_thesis:_for_a_being_Scalar_of_K for_v1_being_Vector_of_V1_holds_(Mx2Tran_(M,b1,B2))_._(a_*_v1)_=_a_*_((Mx2Tran_(M,b1,B2))_._v1) let a be Scalar of K; ::_thesis: for v1 being Vector of V1 holds (Mx2Tran (M,b1,B2)) . (a * v1) = a * ((Mx2Tran (M,b1,B2)) . v1) let v1 be Vector of V1; ::_thesis: (Mx2Tran (M,b1,B2)) . (a * v1) = a * ((Mx2Tran (M,b1,B2)) . v1) thus (Mx2Tran (M,b1,B2)) . (a * v1) = 0. V2 by A1, Th33 .= a * (0. V2) by VECTSP_1:14 .= a * ((Mx2Tran (M,b1,B2)) . v1) by A1, Th33 ; ::_thesis: verum end; now__::_thesis:_for_v1,_w1_being_Vector_of_V1_holds_(Mx2Tran_(M,b1,B2))_._(v1_+_w1)_=_((Mx2Tran_(M,b1,B2))_._v1)_+_((Mx2Tran_(M,b1,B2))_._w1) let v1, w1 be Vector of V1; ::_thesis: (Mx2Tran (M,b1,B2)) . (v1 + w1) = ((Mx2Tran (M,b1,B2)) . v1) + ((Mx2Tran (M,b1,B2)) . w1) thus (Mx2Tran (M,b1,B2)) . (v1 + w1) = 0. V2 by A1, Th33 .= (0. V2) + (0. V2) by RLVECT_1:def_4 .= ((Mx2Tran (M,b1,B2)) . v1) + (0. V2) by A1, Th33 .= ((Mx2Tran (M,b1,B2)) . v1) + ((Mx2Tran (M,b1,B2)) . w1) by A1, Th33 ; ::_thesis: verum end; then ( Mx2Tran (M,b1,B2) is additive & Mx2Tran (M,b1,B2) is homogeneous ) by A2, VECTSP_1:def_20, MOD_2:def_2; hence ( Mx2Tran (M,b1,B2) is homogeneous & Mx2Tran (M,b1,B2) is additive ) ; ::_thesis: verum end; supposeA3: len b1 > 0 ; ::_thesis: ( Mx2Tran (M,b1,B2) is homogeneous & Mx2Tran (M,b1,B2) is additive ) A4: now__::_thesis:_for_v1,_w1_being_Vector_of_V1_holds_(Mx2Tran_(M,b1,B2))_._(v1_+_w1)_=_((Mx2Tran_(M,b1,B2))_._v1)_+_((Mx2Tran_(M,b1,B2))_._w1) let v1, w1 be Vector of V1; ::_thesis: (Mx2Tran (M,b1,B2)) . (v1 + w1) = ((Mx2Tran (M,b1,B2)) . v1) + ((Mx2Tran (M,b1,B2)) . w1) set vb = v1 |-- b1; set wb = w1 |-- b1; set vwb = (v1 + w1) |-- b1; set Lvw = LineVec2Mx ((v1 + w1) |-- b1); set Lv = LineVec2Mx (v1 |-- b1); set Lw = LineVec2Mx (w1 |-- b1); set LLvw = Line (((LineVec2Mx ((v1 + w1) |-- b1)) * M),1); set LLv = Line (((LineVec2Mx (v1 |-- b1)) * M),1); set LLw = Line (((LineVec2Mx (w1 |-- b1)) * M),1); A5: len (LineVec2Mx (w1 |-- b1)) = 1 by MATRIX_1:23; A6: len b1 = len (v1 |-- b1) by MATRLIN:def_7; A7: len M = len b1 by A3, MATRIX_1:23; A8: width (LineVec2Mx (v1 |-- b1)) = len (v1 |-- b1) by MATRIX_1:23; then A9: width ((LineVec2Mx (v1 |-- b1)) * M) = width M by A7, A6, MATRIX_3:def_4; then A10: len (Line (((LineVec2Mx (v1 |-- b1)) * M),1)) = width M by CARD_1:def_7; A11: len (LineVec2Mx (v1 |-- b1)) = 1 by MATRIX_1:23; then A12: len ((LineVec2Mx (v1 |-- b1)) * M) = 1 by A8, A7, A6, MATRIX_3:def_4; A13: len b1 = len (w1 |-- b1) by MATRLIN:def_7; ( width (LineVec2Mx ((v1 + w1) |-- b1)) = len ((v1 + w1) |-- b1) & len b1 = len ((v1 + w1) |-- b1) ) by MATRIX_1:23, MATRLIN:def_7; then width ((LineVec2Mx ((v1 + w1) |-- b1)) * M) = width M by A7, MATRIX_3:def_4; then A14: len (Line (((LineVec2Mx ((v1 + w1) |-- b1)) * M),1)) = width M by CARD_1:def_7; A15: dom (lmlt ((Line (((LineVec2Mx ((v1 + w1) |-- b1)) * M),1)),B2)) = (dom (Line (((LineVec2Mx ((v1 + w1) |-- b1)) * M),1))) /\ (dom B2) by Lm1 .= (dom (Line (((LineVec2Mx (v1 |-- b1)) * M),1))) /\ (dom B2) by A14, A10, FINSEQ_3:29 .= dom (lmlt ((Line (((LineVec2Mx (v1 |-- b1)) * M),1)),B2)) by Lm1 ; A16: width (LineVec2Mx (w1 |-- b1)) = len (w1 |-- b1) by MATRIX_1:23; then A17: width ((LineVec2Mx (w1 |-- b1)) * M) = width M by A7, A13, MATRIX_3:def_4; then A18: len (Line (((LineVec2Mx (w1 |-- b1)) * M),1)) = width M by CARD_1:def_7; A19: dom (lmlt ((Line (((LineVec2Mx ((v1 + w1) |-- b1)) * M),1)),B2)) = (dom (Line (((LineVec2Mx ((v1 + w1) |-- b1)) * M),1))) /\ (dom B2) by Lm1 .= (dom (Line (((LineVec2Mx (w1 |-- b1)) * M),1))) /\ (dom B2) by A14, A18, FINSEQ_3:29 .= dom (lmlt ((Line (((LineVec2Mx (w1 |-- b1)) * M),1)),B2)) by Lm1 ; then A20: len (lmlt ((Line (((LineVec2Mx ((v1 + w1) |-- b1)) * M),1)),B2)) = len (lmlt ((Line (((LineVec2Mx (w1 |-- b1)) * M),1)),B2)) by FINSEQ_3:29; LineVec2Mx ((v1 + w1) |-- b1) = LineVec2Mx ((v1 |-- b1) + (w1 |-- b1)) by Th17 .= (LineVec2Mx (v1 |-- b1)) + (LineVec2Mx (w1 |-- b1)) by A6, A13, MATRIX15:27 ; then (LineVec2Mx ((v1 + w1) |-- b1)) * M = ((LineVec2Mx (v1 |-- b1)) * M) + ((LineVec2Mx (w1 |-- b1)) * M) by A3, A11, A8, A5, A16, A7, A6, A13, MATRIX_4:63; then Line (((LineVec2Mx ((v1 + w1) |-- b1)) * M),1) = (Line (((LineVec2Mx (v1 |-- b1)) * M),1)) + (Line (((LineVec2Mx (w1 |-- b1)) * M),1)) by A12, A9, A17, Lm5; then A21: lmlt ((Line (((LineVec2Mx ((v1 + w1) |-- b1)) * M),1)),B2) = (lmlt ((Line (((LineVec2Mx (v1 |-- b1)) * M),1)),B2)) + (lmlt ((Line (((LineVec2Mx (w1 |-- b1)) * M),1)),B2)) by Th7; A22: now__::_thesis:_for_i_being_Element_of_NAT_st_i_in_dom_(lmlt_((Line_(((LineVec2Mx_(v1_|--_b1))_*_M),1)),B2))_holds_ (lmlt_((Line_(((LineVec2Mx_((v1_+_w1)_|--_b1))_*_M),1)),B2))_._i_=_((lmlt_((Line_(((LineVec2Mx_(v1_|--_b1))_*_M),1)),B2))_/._i)_+_((lmlt_((Line_(((LineVec2Mx_(w1_|--_b1))_*_M),1)),B2))_/._i) let i be Element of NAT ; ::_thesis: ( i in dom (lmlt ((Line (((LineVec2Mx (v1 |-- b1)) * M),1)),B2)) implies (lmlt ((Line (((LineVec2Mx ((v1 + w1) |-- b1)) * M),1)),B2)) . i = ((lmlt ((Line (((LineVec2Mx (v1 |-- b1)) * M),1)),B2)) /. i) + ((lmlt ((Line (((LineVec2Mx (w1 |-- b1)) * M),1)),B2)) /. i) ) assume A23: i in dom (lmlt ((Line (((LineVec2Mx (v1 |-- b1)) * M),1)),B2)) ; ::_thesis: (lmlt ((Line (((LineVec2Mx ((v1 + w1) |-- b1)) * M),1)),B2)) . i = ((lmlt ((Line (((LineVec2Mx (v1 |-- b1)) * M),1)),B2)) /. i) + ((lmlt ((Line (((LineVec2Mx (w1 |-- b1)) * M),1)),B2)) /. i) ( (lmlt ((Line (((LineVec2Mx (v1 |-- b1)) * M),1)),B2)) /. i = (lmlt ((Line (((LineVec2Mx (v1 |-- b1)) * M),1)),B2)) . i & (lmlt ((Line (((LineVec2Mx (w1 |-- b1)) * M),1)),B2)) /. i = (lmlt ((Line (((LineVec2Mx (w1 |-- b1)) * M),1)),B2)) . i ) by A15, A19, A23, PARTFUN1:def_6; hence (lmlt ((Line (((LineVec2Mx ((v1 + w1) |-- b1)) * M),1)),B2)) . i = ((lmlt ((Line (((LineVec2Mx (v1 |-- b1)) * M),1)),B2)) /. i) + ((lmlt ((Line (((LineVec2Mx (w1 |-- b1)) * M),1)),B2)) /. i) by A21, A15, A23, FVSUM_1:17; ::_thesis: verum end; len (lmlt ((Line (((LineVec2Mx ((v1 + w1) |-- b1)) * M),1)),B2)) = len (lmlt ((Line (((LineVec2Mx (v1 |-- b1)) * M),1)),B2)) by A15, FINSEQ_3:29; then Sum (lmlt ((Line (((LineVec2Mx ((v1 + w1) |-- b1)) * M),1)),B2)) = (Sum (lmlt ((Line (((LineVec2Mx (v1 |-- b1)) * M),1)),B2))) + (Sum (lmlt ((Line (((LineVec2Mx (w1 |-- b1)) * M),1)),B2))) by A20, A22, RLVECT_2:2; hence (Mx2Tran (M,b1,B2)) . (v1 + w1) = (Sum (lmlt ((Line (((LineVec2Mx (v1 |-- b1)) * M),1)),B2))) + (Sum (lmlt ((Line (((LineVec2Mx (w1 |-- b1)) * M),1)),B2))) by Def3 .= ((Mx2Tran (M,b1,B2)) . v1) + (Sum (lmlt ((Line (((LineVec2Mx (w1 |-- b1)) * M),1)),B2))) by Def3 .= ((Mx2Tran (M,b1,B2)) . v1) + ((Mx2Tran (M,b1,B2)) . w1) by Def3 ; ::_thesis: verum end; now__::_thesis:_for_a_being_Scalar_of_K for_v1_being_Vector_of_V1_holds_(Mx2Tran_(M,b1,B2))_._(a_*_v1)_=_a_*_((Mx2Tran_(M,b1,B2))_._v1) let a be Scalar of K; ::_thesis: for v1 being Vector of V1 holds (Mx2Tran (M,b1,B2)) . (a * v1) = a * ((Mx2Tran (M,b1,B2)) . v1) let v1 be Vector of V1; ::_thesis: (Mx2Tran (M,b1,B2)) . (a * v1) = a * ((Mx2Tran (M,b1,B2)) . v1) set vb = v1 |-- b1; set avb = (a * v1) |-- b1; set Lav = LineVec2Mx ((a * v1) |-- b1); set Lv = LineVec2Mx (v1 |-- b1); set LLav = Line (((LineVec2Mx ((a * v1) |-- b1)) * M),1); set LLv = Line (((LineVec2Mx (v1 |-- b1)) * M),1); A24: len M = len b1 by A3, MATRIX_1:23; ( width (LineVec2Mx ((a * v1) |-- b1)) = len ((a * v1) |-- b1) & len b1 = len ((a * v1) |-- b1) ) by MATRIX_1:23, MATRLIN:def_7; then width ((LineVec2Mx ((a * v1) |-- b1)) * M) = width M by A24, MATRIX_3:def_4; then A25: len (Line (((LineVec2Mx ((a * v1) |-- b1)) * M),1)) = width M by CARD_1:def_7; A26: ( width (LineVec2Mx (v1 |-- b1)) = len (v1 |-- b1) & len b1 = len (v1 |-- b1) ) by MATRIX_1:23, MATRLIN:def_7; then width ((LineVec2Mx (v1 |-- b1)) * M) = width M by A24, MATRIX_3:def_4; then len (Line (((LineVec2Mx (v1 |-- b1)) * M),1)) = width M by CARD_1:def_7; then A27: dom (Line (((LineVec2Mx ((a * v1) |-- b1)) * M),1)) = dom (Line (((LineVec2Mx (v1 |-- b1)) * M),1)) by A25, FINSEQ_3:29; LineVec2Mx ((a * v1) |-- b1) = LineVec2Mx (a * (v1 |-- b1)) by Th18 .= a * (LineVec2Mx (v1 |-- b1)) by MATRIX15:29 ; then A28: (LineVec2Mx ((a * v1) |-- b1)) * M = a * ((LineVec2Mx (v1 |-- b1)) * M) by A24, A26, MATRIX15:1; A29: dom (lmlt ((Line (((LineVec2Mx ((a * v1) |-- b1)) * M),1)),B2)) = (dom (Line (((LineVec2Mx ((a * v1) |-- b1)) * M),1))) /\ (dom B2) by Lm1; A30: (dom (Line (((LineVec2Mx (v1 |-- b1)) * M),1))) /\ (dom B2) = dom (lmlt ((Line (((LineVec2Mx (v1 |-- b1)) * M),1)),B2)) by Lm1; len (LineVec2Mx (v1 |-- b1)) = 1 by MATRIX_1:23; then len ((LineVec2Mx (v1 |-- b1)) * M) = 1 by A24, A26, MATRIX_3:def_4; then A31: Line (((LineVec2Mx ((a * v1) |-- b1)) * M),1) = a * (Line (((LineVec2Mx (v1 |-- b1)) * M),1)) by A28, MATRIXR1:20; A32: now__::_thesis:_for_i_being_Element_of_NAT_st_i_in_dom_(lmlt_((Line_(((LineVec2Mx_(v1_|--_b1))_*_M),1)),B2))_holds_ (lmlt_((Line_(((LineVec2Mx_((a_*_v1)_|--_b1))_*_M),1)),B2))_._i_=_a_*_((lmlt_((Line_(((LineVec2Mx_(v1_|--_b1))_*_M),1)),B2))_/._i) let i be Element of NAT ; ::_thesis: ( i in dom (lmlt ((Line (((LineVec2Mx (v1 |-- b1)) * M),1)),B2)) implies (lmlt ((Line (((LineVec2Mx ((a * v1) |-- b1)) * M),1)),B2)) . i = a * ((lmlt ((Line (((LineVec2Mx (v1 |-- b1)) * M),1)),B2)) /. i) ) assume A33: i in dom (lmlt ((Line (((LineVec2Mx (v1 |-- b1)) * M),1)),B2)) ; ::_thesis: (lmlt ((Line (((LineVec2Mx ((a * v1) |-- b1)) * M),1)),B2)) . i = a * ((lmlt ((Line (((LineVec2Mx (v1 |-- b1)) * M),1)),B2)) /. i) A34: (lmlt ((Line (((LineVec2Mx (v1 |-- b1)) * M),1)),B2)) . i = (lmlt ((Line (((LineVec2Mx (v1 |-- b1)) * M),1)),B2)) /. i by A33, PARTFUN1:def_6; i in dom (Line (((LineVec2Mx (v1 |-- b1)) * M),1)) by A30, A33, XBOOLE_0:def_4; then A35: (Line (((LineVec2Mx (v1 |-- b1)) * M),1)) . i = (Line (((LineVec2Mx (v1 |-- b1)) * M),1)) /. i by PARTFUN1:def_6; i in dom B2 by A30, A33, XBOOLE_0:def_4; then A36: B2 . i = B2 /. i by PARTFUN1:def_6; A37: i in dom (Line (((LineVec2Mx ((a * v1) |-- b1)) * M),1)) by A27, A30, A33, XBOOLE_0:def_4; then A38: (Line (((LineVec2Mx ((a * v1) |-- b1)) * M),1)) . i = (Line (((LineVec2Mx ((a * v1) |-- b1)) * M),1)) /. i by PARTFUN1:def_6; hence (lmlt ((Line (((LineVec2Mx ((a * v1) |-- b1)) * M),1)),B2)) . i = the lmult of V2 . (((Line (((LineVec2Mx ((a * v1) |-- b1)) * M),1)) /. i),(B2 /. i)) by A29, A27, A30, A33, A36, FUNCOP_1:22 .= (a * ((Line (((LineVec2Mx (v1 |-- b1)) * M),1)) /. i)) * (B2 /. i) by A31, A37, A35, A38, FVSUM_1:50 .= a * (((Line (((LineVec2Mx (v1 |-- b1)) * M),1)) /. i) * (B2 /. i)) by VECTSP_1:def_16 .= a * ((lmlt ((Line (((LineVec2Mx (v1 |-- b1)) * M),1)),B2)) /. i) by A33, A35, A36, A34, FUNCOP_1:22 ; ::_thesis: verum end; len (lmlt ((Line (((LineVec2Mx ((a * v1) |-- b1)) * M),1)),B2)) = len (lmlt ((Line (((LineVec2Mx (v1 |-- b1)) * M),1)),B2)) by A29, A27, A30, FINSEQ_3:29; then Sum (lmlt ((Line (((LineVec2Mx ((a * v1) |-- b1)) * M),1)),B2)) = a * (Sum (lmlt ((Line (((LineVec2Mx (v1 |-- b1)) * M),1)),B2))) by A32, RLVECT_2:67; hence (Mx2Tran (M,b1,B2)) . (a * v1) = a * (Sum (lmlt ((Line (((LineVec2Mx (v1 |-- b1)) * M),1)),B2))) by Def3 .= a * ((Mx2Tran (M,b1,B2)) . v1) by Def3 ; ::_thesis: verum end; then ( Mx2Tran (M,b1,B2) is additive & Mx2Tran (M,b1,B2) is homogeneous ) by A4, VECTSP_1:def_20, MOD_2:def_2; hence ( Mx2Tran (M,b1,B2) is homogeneous & Mx2Tran (M,b1,B2) is additive ) ; ::_thesis: verum end; end; end; end; theorem Th34: :: MATRLIN2:34 for K being Field for V1, V2 being finite-dimensional VectSp of K for f being Function of V1,V2 for b1 being OrdBasis of V1 for b2 being OrdBasis of V2 st f is additive & f is homogeneous holds Mx2Tran ((AutMt (f,b1,b2)),b1,b2) = f proof let K be Field; ::_thesis: for V1, V2 being finite-dimensional VectSp of K for f being Function of V1,V2 for b1 being OrdBasis of V1 for b2 being OrdBasis of V2 st f is additive & f is homogeneous holds Mx2Tran ((AutMt (f,b1,b2)),b1,b2) = f let V1, V2 be finite-dimensional VectSp of K; ::_thesis: for f being Function of V1,V2 for b1 being OrdBasis of V1 for b2 being OrdBasis of V2 st f is additive & f is homogeneous holds Mx2Tran ((AutMt (f,b1,b2)),b1,b2) = f let f be Function of V1,V2; ::_thesis: for b1 being OrdBasis of V1 for b2 being OrdBasis of V2 st f is additive & f is homogeneous holds Mx2Tran ((AutMt (f,b1,b2)),b1,b2) = f let b1 be OrdBasis of V1; ::_thesis: for b2 being OrdBasis of V2 st f is additive & f is homogeneous holds Mx2Tran ((AutMt (f,b1,b2)),b1,b2) = f let b2 be OrdBasis of V2; ::_thesis: ( f is additive & f is homogeneous implies Mx2Tran ((AutMt (f,b1,b2)),b1,b2) = f ) assume A1: ( f is additive & f is homogeneous ) ; ::_thesis: Mx2Tran ((AutMt (f,b1,b2)),b1,b2) = f set A = AutMt (f,b1,b2); set M = Mx2Tran ((AutMt (f,b1,b2)),b1,b2); percases ( len b1 = 0 or len b1 > 0 ) ; supposeA2: len b1 = 0 ; ::_thesis: Mx2Tran ((AutMt (f,b1,b2)),b1,b2) = f now__::_thesis:_for_x_being_set_st_x_in_the_carrier_of_V1_holds_ f_._x_=_(Mx2Tran_((AutMt_(f,b1,b2)),b1,b2))_._x A3: b1 is one-to-one by MATRLIN:def_2; reconsider R = rng b1 as Basis of V1 by MATRLIN:def_2; let x be set ; ::_thesis: ( x in the carrier of V1 implies f . x = (Mx2Tran ((AutMt (f,b1,b2)),b1,b2)) . x ) assume A4: x in the carrier of V1 ; ::_thesis: f . x = (Mx2Tran ((AutMt (f,b1,b2)),b1,b2)) . x A5: Seg (len b1) = {} by A2; dim V1 = card R by VECTSP_9:def_1 .= card (dom b1) by A3, CARD_1:70 .= 0 by A5, CARD_1:27, FINSEQ_1:def_3 ; then (Omega). V1 = (0). V1 by VECTSP_9:29; then the carrier of V1 = {(0. V1)} by VECTSP_4:def_3; then x = 0. V1 by A4, TARSKI:def_1; hence f . x = f . ((0. K) * (0. V1)) by VECTSP_1:15 .= (0. K) * (f . (0. V1)) by A1, MOD_2:def_2 .= 0. V2 by VECTSP_1:15 .= (Mx2Tran ((AutMt (f,b1,b2)),b1,b2)) . x by A2, A4, Th33 ; ::_thesis: verum end; hence Mx2Tran ((AutMt (f,b1,b2)),b1,b2) = f by FUNCT_2:12; ::_thesis: verum end; supposeA6: len b1 > 0 ; ::_thesis: Mx2Tran ((AutMt (f,b1,b2)),b1,b2) = f reconsider fb = f * b1, Mf = (Mx2Tran ((AutMt (f,b1,b2)),b1,b2)) * b1 as FinSequence ; A7: rng b1 is Subset of V1 by FINSEQ_1:def_4; dom f = the carrier of V1 by FUNCT_2:def_1; then A8: len fb = len b1 by A7, FINSEQ_2:29; dom (Mx2Tran ((AutMt (f,b1,b2)),b1,b2)) = the carrier of V1 by FUNCT_2:def_1; then A9: len Mf = len b1 by A7, FINSEQ_2:29; now__::_thesis:_for_i_being_Nat_st_1_<=_i_&_i_<=_len_fb_holds_ fb_._i_=_Mf_._i A10: dom fb = dom Mf by A8, A9, FINSEQ_3:29; let i be Nat; ::_thesis: ( 1 <= i & i <= len fb implies fb . i = Mf . i ) assume ( 1 <= i & i <= len fb ) ; ::_thesis: fb . i = Mf . i then A11: i in dom fb by FINSEQ_3:25; dom fb = dom b1 by A8, FINSEQ_3:29; then A12: b1 . i = b1 /. i by A11, PARTFUN1:def_6; LineVec2Mx (((Mx2Tran ((AutMt (f,b1,b2)),b1,b2)) . (b1 /. i)) |-- b2) = (LineVec2Mx ((b1 /. i) |-- b1)) * (AutMt (f,b1,b2)) by A6, Th32 .= LineVec2Mx ((f . (b1 /. i)) |-- b2) by A1, A6, Th31 ; then ((Mx2Tran ((AutMt (f,b1,b2)),b1,b2)) . (b1 /. i)) |-- b2 = Line ((LineVec2Mx ((f . (b1 /. i)) |-- b2)),1) by MATRIX15:25 .= (f . (b1 /. i)) |-- b2 by MATRIX15:25 ; then (Mx2Tran ((AutMt (f,b1,b2)),b1,b2)) . (b1 /. i) = f . (b1 /. i) by MATRLIN:34; hence fb . i = (Mx2Tran ((AutMt (f,b1,b2)),b1,b2)) . (b1 /. i) by A11, A12, FUNCT_1:12 .= Mf . i by A11, A10, A12, FUNCT_1:12 ; ::_thesis: verum end; hence Mx2Tran ((AutMt (f,b1,b2)),b1,b2) = f by A1, A6, A8, A9, FINSEQ_1:14, MATRLIN:22; ::_thesis: verum end; end; end; theorem Th35: :: MATRLIN2:35 for i being Nat for K being Field for A, B being Matrix of K st i in dom A & width A = len B holds (LineVec2Mx (Line (A,i))) * B = LineVec2Mx (Line ((A * B),i)) proof let i be Nat; ::_thesis: for K being Field for A, B being Matrix of K st i in dom A & width A = len B holds (LineVec2Mx (Line (A,i))) * B = LineVec2Mx (Line ((A * B),i)) let K be Field; ::_thesis: for A, B being Matrix of K st i in dom A & width A = len B holds (LineVec2Mx (Line (A,i))) * B = LineVec2Mx (Line ((A * B),i)) let A, B be Matrix of K; ::_thesis: ( i in dom A & width A = len B implies (LineVec2Mx (Line (A,i))) * B = LineVec2Mx (Line ((A * B),i)) ) assume that A1: i in dom A and A2: width A = len B ; ::_thesis: (LineVec2Mx (Line (A,i))) * B = LineVec2Mx (Line ((A * B),i)) A3: width (A * B) = width B by A2, MATRIX_3:def_4; set LAB = LineVec2Mx (Line ((A * B),i)); A4: ( width (LineVec2Mx (Line ((A * B),i))) = len (Line ((A * B),i)) & len (Line ((A * B),i)) = width (A * B) ) by CARD_1:def_7, MATRIX_1:23; set L = LineVec2Mx (Line (A,i)); A5: ( width (LineVec2Mx (Line (A,i))) = len (Line (A,i)) & len (Line (A,i)) = width A ) by CARD_1:def_7, MATRIX_1:23; then A6: width ((LineVec2Mx (Line (A,i))) * B) = width B by A2, MATRIX_3:def_4; len (LineVec2Mx (Line (A,i))) = 1 by CARD_1:def_7; then A7: len ((LineVec2Mx (Line (A,i))) * B) = 1 by A2, A5, MATRIX_3:def_4; len (A * B) = len A by A2, MATRIX_3:def_4; then A8: dom A = dom (A * B) by FINSEQ_3:29; A9: now__::_thesis:_for_j,_k_being_Nat_st_[j,k]_in_Indices_((LineVec2Mx_(Line_(A,i)))_*_B)_holds_ ((LineVec2Mx_(Line_(A,i)))_*_B)_*_(j,k)_=_(LineVec2Mx_(Line_((A_*_B),i)))_*_(j,k) let j, k be Nat; ::_thesis: ( [j,k] in Indices ((LineVec2Mx (Line (A,i))) * B) implies ((LineVec2Mx (Line (A,i))) * B) * (j,k) = (LineVec2Mx (Line ((A * B),i))) * (j,k) ) assume A10: [j,k] in Indices ((LineVec2Mx (Line (A,i))) * B) ; ::_thesis: ((LineVec2Mx (Line (A,i))) * B) * (j,k) = (LineVec2Mx (Line ((A * B),i))) * (j,k) A11: k in Seg (width (A * B)) by A3, A6, A10, ZFMISC_1:87; then A12: [i,k] in Indices (A * B) by A1, A8, ZFMISC_1:87; Indices ((LineVec2Mx (Line (A,i))) * B) = [:(Seg 1),(Seg (width B)):] by A7, A6, FINSEQ_1:def_3; then j in Seg 1 by A10, ZFMISC_1:87; then A13: j = 1 by FINSEQ_1:2, TARSKI:def_1; hence ((LineVec2Mx (Line (A,i))) * B) * (j,k) = (Line ((LineVec2Mx (Line (A,i))),1)) "*" (Col (B,k)) by A2, A5, A10, MATRIX_3:def_4 .= (Line (A,i)) "*" (Col (B,k)) by MATRIX15:25 .= (A * B) * (i,k) by A2, A12, MATRIX_3:def_4 .= (Line ((A * B),i)) . k by A11, MATRIX_1:def_7 .= (Line ((LineVec2Mx (Line ((A * B),i))),j)) . k by A13, MATRIX15:25 .= (LineVec2Mx (Line ((A * B),i))) * (j,k) by A4, A11, MATRIX_1:def_7 ; ::_thesis: verum end; len (LineVec2Mx (Line ((A * B),i))) = 1 by CARD_1:def_7; hence (LineVec2Mx (Line (A,i))) * B = LineVec2Mx (Line ((A * B),i)) by A4, A3, A7, A6, A9, MATRIX_1:21; ::_thesis: verum end; theorem Th36: :: MATRLIN2:36 for K being Field for V1, V2 being finite-dimensional VectSp of K for b1 being OrdBasis of V1 for b2 being OrdBasis of V2 for M being Matrix of len b1, len b2,K holds AutMt ((Mx2Tran (M,b1,b2)),b1,b2) = M proof let K be Field; ::_thesis: for V1, V2 being finite-dimensional VectSp of K for b1 being OrdBasis of V1 for b2 being OrdBasis of V2 for M being Matrix of len b1, len b2,K holds AutMt ((Mx2Tran (M,b1,b2)),b1,b2) = M let V1, V2 be finite-dimensional VectSp of K; ::_thesis: for b1 being OrdBasis of V1 for b2 being OrdBasis of V2 for M being Matrix of len b1, len b2,K holds AutMt ((Mx2Tran (M,b1,b2)),b1,b2) = M let b1 be OrdBasis of V1; ::_thesis: for b2 being OrdBasis of V2 for M being Matrix of len b1, len b2,K holds AutMt ((Mx2Tran (M,b1,b2)),b1,b2) = M let b2 be OrdBasis of V2; ::_thesis: for M being Matrix of len b1, len b2,K holds AutMt ((Mx2Tran (M,b1,b2)),b1,b2) = M let M be Matrix of len b1, len b2,K; ::_thesis: AutMt ((Mx2Tran (M,b1,b2)),b1,b2) = M set MX = Mx2Tran (M,b1,b2); set A = AutMt ((Mx2Tran (M,b1,b2)),b1,b2); set ONE = 1. (K,(len b1)); A1: len M = len b1 by MATRIX_1:25; A2: len (AutMt ((Mx2Tran (M,b1,b2)),b1,b2)) = len b1 by MATRIX_1:25; A3: len (1. (K,(len b1))) = len b1 by MATRIX_1:24; now__::_thesis:_for_i_being_Nat_st_1_<=_i_&_i_<=_len_M_holds_ M_._i_=_(AutMt_((Mx2Tran_(M,b1,b2)),b1,b2))_._i let i be Nat; ::_thesis: ( 1 <= i & i <= len M implies M . i = (AutMt ((Mx2Tran (M,b1,b2)),b1,b2)) . i ) assume A4: ( 1 <= i & i <= len M ) ; ::_thesis: M . i = (AutMt ((Mx2Tran (M,b1,b2)),b1,b2)) . i A5: i in Seg (len b1) by A1, A4, FINSEQ_1:1; A6: i in dom (1. (K,(len b1))) by A1, A3, A4, FINSEQ_3:25; reconsider Ai = (AutMt ((Mx2Tran (M,b1,b2)),b1,b2)) /. i as FinSequence of K by FINSEQ_1:def_11; A7: i in dom b1 by A1, A4, FINSEQ_3:25; then (AutMt ((Mx2Tran (M,b1,b2)),b1,b2)) /. i = ((Mx2Tran (M,b1,b2)) . (b1 /. i)) |-- b2 by MATRLIN:def_8; then LineVec2Mx Ai = (LineVec2Mx ((b1 /. i) |-- b1)) * M by A1, A4, Th32 .= (LineVec2Mx (Line ((1. (K,(len b1))),i))) * M by A7, Th19 .= LineVec2Mx (Line (((1. (K,(len b1))) * M),i)) by A1, A6, Th35, MATRIX_1:24 .= LineVec2Mx (Line (M,i)) by A1, MATRIXR2:68 ; then A8: Ai = Line ((LineVec2Mx (Line (M,i))),1) by MATRIX15:25 .= Line (M,i) by MATRIX15:25 .= M . i by A5, MATRIX_2:8 ; i in dom (AutMt ((Mx2Tran (M,b1,b2)),b1,b2)) by A1, A2, A4, FINSEQ_3:25; hence M . i = (AutMt ((Mx2Tran (M,b1,b2)),b1,b2)) . i by A8, PARTFUN1:def_6; ::_thesis: verum end; hence AutMt ((Mx2Tran (M,b1,b2)),b1,b2) = M by A2, FINSEQ_1:14, MATRIX_1:25; ::_thesis: verum end; definition let n, m be Nat; let K be Field; let A be Matrix of n,m,K; let B be Matrix of K; :: original: + redefine funcA + B -> Matrix of n,m,K; coherence A + B is Matrix of n,m,K proof A1: ( width (A + B) = width A & len A = n ) by MATRIX_1:def_2, MATRIX_3:def_3; A2: len (A + B) = len A by MATRIX_3:def_3; percases ( n = 0 or n > 0 ) ; supposeA3: n = 0 ; ::_thesis: A + B is Matrix of n,m,K then A + B = {} by A2, MATRIX_1:def_2; hence A + B is Matrix of n,m,K by A3, MATRIX_1:13; ::_thesis: verum end; suppose n > 0 ; ::_thesis: A + B is Matrix of n,m,K then width A = m by MATRIX_1:23; hence A + B is Matrix of n,m,K by A2, A1, MATRIX_2:7; ::_thesis: verum end; end; end; end; theorem :: MATRLIN2:37 for K being Field for V1, V2 being finite-dimensional VectSp of K for b1 being OrdBasis of V1 for B2 being FinSequence of V2 for A, B being Matrix of len b1, len B2,K holds Mx2Tran ((A + B),b1,B2) = (Mx2Tran (A,b1,B2)) + (Mx2Tran (B,b1,B2)) proof let K be Field; ::_thesis: for V1, V2 being finite-dimensional VectSp of K for b1 being OrdBasis of V1 for B2 being FinSequence of V2 for A, B being Matrix of len b1, len B2,K holds Mx2Tran ((A + B),b1,B2) = (Mx2Tran (A,b1,B2)) + (Mx2Tran (B,b1,B2)) let V1, V2 be finite-dimensional VectSp of K; ::_thesis: for b1 being OrdBasis of V1 for B2 being FinSequence of V2 for A, B being Matrix of len b1, len B2,K holds Mx2Tran ((A + B),b1,B2) = (Mx2Tran (A,b1,B2)) + (Mx2Tran (B,b1,B2)) let b1 be OrdBasis of V1; ::_thesis: for B2 being FinSequence of V2 for A, B being Matrix of len b1, len B2,K holds Mx2Tran ((A + B),b1,B2) = (Mx2Tran (A,b1,B2)) + (Mx2Tran (B,b1,B2)) let B2 be FinSequence of V2; ::_thesis: for A, B being Matrix of len b1, len B2,K holds Mx2Tran ((A + B),b1,B2) = (Mx2Tran (A,b1,B2)) + (Mx2Tran (B,b1,B2)) let A, B be Matrix of len b1, len B2,K; ::_thesis: Mx2Tran ((A + B),b1,B2) = (Mx2Tran (A,b1,B2)) + (Mx2Tran (B,b1,B2)) set AB = A + B; set M = Mx2Tran ((A + B),b1,B2); set MA = Mx2Tran (A,b1,B2); set MB = Mx2Tran (B,b1,B2); now__::_thesis:_for_x_being_set_st_x_in_the_carrier_of_V1_holds_ (Mx2Tran_((A_+_B),b1,B2))_._x_=_((Mx2Tran_(A,b1,B2))_+_(Mx2Tran_(B,b1,B2)))_._x let x be set ; ::_thesis: ( x in the carrier of V1 implies (Mx2Tran ((A + B),b1,B2)) . x = ((Mx2Tran (A,b1,B2)) + (Mx2Tran (B,b1,B2))) . x ) assume A1: x in the carrier of V1 ; ::_thesis: (Mx2Tran ((A + B),b1,B2)) . x = ((Mx2Tran (A,b1,B2)) + (Mx2Tran (B,b1,B2))) . x reconsider v = x as Element of V1 by A1; now__::_thesis:_(Mx2Tran_((A_+_B),b1,B2))_._x_=_((Mx2Tran_(A,b1,B2))_+_(Mx2Tran_(B,b1,B2)))_._x percases ( len b1 = 0 or len b1 > 0 ) ; supposeA2: len b1 = 0 ; ::_thesis: (Mx2Tran ((A + B),b1,B2)) . x = ((Mx2Tran (A,b1,B2)) + (Mx2Tran (B,b1,B2))) . x hence (Mx2Tran ((A + B),b1,B2)) . x = 0. V2 by A1, Th33 .= (0. V2) + (0. V2) by RLVECT_1:def_4 .= ((Mx2Tran (A,b1,B2)) . v) + (0. V2) by A2, Th33 .= ((Mx2Tran (A,b1,B2)) . v) + ((Mx2Tran (B,b1,B2)) . v) by A2, Th33 .= ((Mx2Tran (A,b1,B2)) + (Mx2Tran (B,b1,B2))) . x by MATRLIN:def_3 ; ::_thesis: verum end; supposeA3: len b1 > 0 ; ::_thesis: (Mx2Tran ((A + B),b1,B2)) . x = ((Mx2Tran (A,b1,B2)) + (Mx2Tran (B,b1,B2))) . x set L = LineVec2Mx (v |-- b1); A4: ( width (LineVec2Mx (v |-- b1)) = len (v |-- b1) & len (v |-- b1) = len b1 ) by MATRIX_1:23, MATRLIN:def_7; set mB = lmlt ((Line (((LineVec2Mx (v |-- b1)) * B),1)),B2); A5: ( len B = len b1 & width B = len B2 ) by A3, MATRIX_1:23; then A6: width ((LineVec2Mx (v |-- b1)) * B) = len B2 by A4, MATRIX_3:def_4; then len (Line (((LineVec2Mx (v |-- b1)) * B),1)) = len B2 by CARD_1:def_7; then dom (Line (((LineVec2Mx (v |-- b1)) * B),1)) = dom B2 by FINSEQ_3:29; then A7: dom (lmlt ((Line (((LineVec2Mx (v |-- b1)) * B),1)),B2)) = dom B2 by MATRLIN:12; then A8: len (lmlt ((Line (((LineVec2Mx (v |-- b1)) * B),1)),B2)) = len B2 by FINSEQ_3:29; A9: len A = len b1 by A3, MATRIX_1:23; A10: len (LineVec2Mx (v |-- b1)) = 1 by MATRIX_1:23; then A11: len ((LineVec2Mx (v |-- b1)) * A) = 1 by A9, A4, MATRIX_3:def_4; set mA = lmlt ((Line (((LineVec2Mx (v |-- b1)) * A),1)),B2); A12: width A = len B2 by A3, MATRIX_1:23; then A13: width ((LineVec2Mx (v |-- b1)) * A) = len B2 by A9, A4, MATRIX_3:def_4; then len (Line (((LineVec2Mx (v |-- b1)) * A),1)) = len B2 by CARD_1:def_7; then dom (Line (((LineVec2Mx (v |-- b1)) * A),1)) = dom B2 by FINSEQ_3:29; then A14: dom (lmlt ((Line (((LineVec2Mx (v |-- b1)) * A),1)),B2)) = dom B2 by MATRLIN:12; then A15: len (lmlt ((Line (((LineVec2Mx (v |-- b1)) * A),1)),B2)) = len B2 by FINSEQ_3:29; A16: dom ((lmlt ((Line (((LineVec2Mx (v |-- b1)) * A),1)),B2)) + (lmlt ((Line (((LineVec2Mx (v |-- b1)) * B),1)),B2))) = (dom B2) /\ (dom B2) by A14, A7, Lm3 .= dom B2 ; then A17: len ((lmlt ((Line (((LineVec2Mx (v |-- b1)) * A),1)),B2)) + (lmlt ((Line (((LineVec2Mx (v |-- b1)) * B),1)),B2))) = len B2 by FINSEQ_3:29; A18: now__::_thesis:_for_k_being_Element_of_NAT_st_k_in_dom_(lmlt_((Line_(((LineVec2Mx_(v_|--_b1))_*_A),1)),B2))_holds_ ((lmlt_((Line_(((LineVec2Mx_(v_|--_b1))_*_A),1)),B2))_+_(lmlt_((Line_(((LineVec2Mx_(v_|--_b1))_*_B),1)),B2)))_._k_=_((lmlt_((Line_(((LineVec2Mx_(v_|--_b1))_*_A),1)),B2))_/._k)_+_((lmlt_((Line_(((LineVec2Mx_(v_|--_b1))_*_B),1)),B2))_/._k) let k be Element of NAT ; ::_thesis: ( k in dom (lmlt ((Line (((LineVec2Mx (v |-- b1)) * A),1)),B2)) implies ((lmlt ((Line (((LineVec2Mx (v |-- b1)) * A),1)),B2)) + (lmlt ((Line (((LineVec2Mx (v |-- b1)) * B),1)),B2))) . k = ((lmlt ((Line (((LineVec2Mx (v |-- b1)) * A),1)),B2)) /. k) + ((lmlt ((Line (((LineVec2Mx (v |-- b1)) * B),1)),B2)) /. k) ) assume A19: k in dom (lmlt ((Line (((LineVec2Mx (v |-- b1)) * A),1)),B2)) ; ::_thesis: ((lmlt ((Line (((LineVec2Mx (v |-- b1)) * A),1)),B2)) + (lmlt ((Line (((LineVec2Mx (v |-- b1)) * B),1)),B2))) . k = ((lmlt ((Line (((LineVec2Mx (v |-- b1)) * A),1)),B2)) /. k) + ((lmlt ((Line (((LineVec2Mx (v |-- b1)) * B),1)),B2)) /. k) ( (lmlt ((Line (((LineVec2Mx (v |-- b1)) * A),1)),B2)) /. k = (lmlt ((Line (((LineVec2Mx (v |-- b1)) * A),1)),B2)) . k & (lmlt ((Line (((LineVec2Mx (v |-- b1)) * B),1)),B2)) /. k = (lmlt ((Line (((LineVec2Mx (v |-- b1)) * B),1)),B2)) . k ) by A14, A7, A19, PARTFUN1:def_6; hence ((lmlt ((Line (((LineVec2Mx (v |-- b1)) * A),1)),B2)) + (lmlt ((Line (((LineVec2Mx (v |-- b1)) * B),1)),B2))) . k = ((lmlt ((Line (((LineVec2Mx (v |-- b1)) * A),1)),B2)) /. k) + ((lmlt ((Line (((LineVec2Mx (v |-- b1)) * B),1)),B2)) /. k) by A14, A16, A19, FVSUM_1:17; ::_thesis: verum end; thus (Mx2Tran ((A + B),b1,B2)) . x = Sum (lmlt ((Line (((LineVec2Mx (v |-- b1)) * (A + B)),1)),B2)) by Def3 .= Sum (lmlt ((Line ((((LineVec2Mx (v |-- b1)) * A) + ((LineVec2Mx (v |-- b1)) * B)),1)),B2)) by A3, A10, A9, A12, A5, A4, MATRIX_4:62 .= Sum (lmlt (((Line (((LineVec2Mx (v |-- b1)) * A),1)) + (Line (((LineVec2Mx (v |-- b1)) * B),1))),B2)) by A11, A13, A6, Lm5 .= Sum ((lmlt ((Line (((LineVec2Mx (v |-- b1)) * A),1)),B2)) + (lmlt ((Line (((LineVec2Mx (v |-- b1)) * B),1)),B2))) by Th7 .= (Sum (lmlt ((Line (((LineVec2Mx (v |-- b1)) * A),1)),B2))) + (Sum (lmlt ((Line (((LineVec2Mx (v |-- b1)) * B),1)),B2))) by A15, A8, A17, A18, RLVECT_2:2 .= ((Mx2Tran (A,b1,B2)) . v) + (Sum (lmlt ((Line (((LineVec2Mx (v |-- b1)) * B),1)),B2))) by Def3 .= ((Mx2Tran (A,b1,B2)) . v) + ((Mx2Tran (B,b1,B2)) . v) by Def3 .= ((Mx2Tran (A,b1,B2)) + (Mx2Tran (B,b1,B2))) . x by MATRLIN:def_3 ; ::_thesis: verum end; end; end; hence (Mx2Tran ((A + B),b1,B2)) . x = ((Mx2Tran (A,b1,B2)) + (Mx2Tran (B,b1,B2))) . x ; ::_thesis: verum end; hence Mx2Tran ((A + B),b1,B2) = (Mx2Tran (A,b1,B2)) + (Mx2Tran (B,b1,B2)) by FUNCT_2:12; ::_thesis: verum end; theorem :: MATRLIN2:38 for K being Field for a being Element of K for V1, V2 being finite-dimensional VectSp of K for b1 being OrdBasis of V1 for B2 being FinSequence of V2 for A being Matrix of len b1, len B2,K holds a * (Mx2Tran (A,b1,B2)) = Mx2Tran ((a * A),b1,B2) proof let K be Field; ::_thesis: for a being Element of K for V1, V2 being finite-dimensional VectSp of K for b1 being OrdBasis of V1 for B2 being FinSequence of V2 for A being Matrix of len b1, len B2,K holds a * (Mx2Tran (A,b1,B2)) = Mx2Tran ((a * A),b1,B2) let a be Element of K; ::_thesis: for V1, V2 being finite-dimensional VectSp of K for b1 being OrdBasis of V1 for B2 being FinSequence of V2 for A being Matrix of len b1, len B2,K holds a * (Mx2Tran (A,b1,B2)) = Mx2Tran ((a * A),b1,B2) let V1, V2 be finite-dimensional VectSp of K; ::_thesis: for b1 being OrdBasis of V1 for B2 being FinSequence of V2 for A being Matrix of len b1, len B2,K holds a * (Mx2Tran (A,b1,B2)) = Mx2Tran ((a * A),b1,B2) let b1 be OrdBasis of V1; ::_thesis: for B2 being FinSequence of V2 for A being Matrix of len b1, len B2,K holds a * (Mx2Tran (A,b1,B2)) = Mx2Tran ((a * A),b1,B2) let B2 be FinSequence of V2; ::_thesis: for A being Matrix of len b1, len B2,K holds a * (Mx2Tran (A,b1,B2)) = Mx2Tran ((a * A),b1,B2) let A be Matrix of len b1, len B2,K; ::_thesis: a * (Mx2Tran (A,b1,B2)) = Mx2Tran ((a * A),b1,B2) set aA = a * A; set aM = Mx2Tran ((a * A),b1,B2); set M = Mx2Tran (A,b1,B2); now__::_thesis:_for_x_being_set_st_x_in_the_carrier_of_V1_holds_ (Mx2Tran_((a_*_A),b1,B2))_._x_=_(a_*_(Mx2Tran_(A,b1,B2)))_._x let x be set ; ::_thesis: ( x in the carrier of V1 implies (Mx2Tran ((a * A),b1,B2)) . x = (a * (Mx2Tran (A,b1,B2))) . x ) assume x in the carrier of V1 ; ::_thesis: (Mx2Tran ((a * A),b1,B2)) . x = (a * (Mx2Tran (A,b1,B2))) . x then reconsider v = x as Element of V1 ; set L = LineVec2Mx (v |-- b1); set amA = lmlt ((a * (Line (((LineVec2Mx (v |-- b1)) * A),1))),B2); set mA = lmlt ((Line (((LineVec2Mx (v |-- b1)) * A),1)),B2); A1: ( width (LineVec2Mx (v |-- b1)) = len (v |-- b1) & len (v |-- b1) = len b1 ) by MATRIX_1:23, MATRLIN:def_7; A2: len A = len b1 by MATRIX_1:def_2; len (LineVec2Mx (v |-- b1)) = 1 by MATRIX_1:23; then A3: len ((LineVec2Mx (v |-- b1)) * A) = 1 by A1, A2, MATRIX_3:def_4; A4: dom (lmlt ((Line (((LineVec2Mx (v |-- b1)) * A),1)),B2)) = (dom (Line (((LineVec2Mx (v |-- b1)) * A),1))) /\ (dom B2) by Lm1; len (a * (Line (((LineVec2Mx (v |-- b1)) * A),1))) = len (Line (((LineVec2Mx (v |-- b1)) * A),1)) by MATRIXR1:16; then A5: dom (a * (Line (((LineVec2Mx (v |-- b1)) * A),1))) = dom (Line (((LineVec2Mx (v |-- b1)) * A),1)) by FINSEQ_3:29; A6: dom (lmlt ((a * (Line (((LineVec2Mx (v |-- b1)) * A),1))),B2)) = (dom (a * (Line (((LineVec2Mx (v |-- b1)) * A),1)))) /\ (dom B2) by Lm1; then A7: len (lmlt ((Line (((LineVec2Mx (v |-- b1)) * A),1)),B2)) = len (lmlt ((a * (Line (((LineVec2Mx (v |-- b1)) * A),1))),B2)) by A5, A4, FINSEQ_3:29; A8: now__::_thesis:_for_k_being_Element_of_NAT_st_k_in_dom_(lmlt_((Line_(((LineVec2Mx_(v_|--_b1))_*_A),1)),B2))_holds_ (lmlt_((a_*_(Line_(((LineVec2Mx_(v_|--_b1))_*_A),1))),B2))_._k_=_a_*_((lmlt_((Line_(((LineVec2Mx_(v_|--_b1))_*_A),1)),B2))_/._k) let k be Element of NAT ; ::_thesis: ( k in dom (lmlt ((Line (((LineVec2Mx (v |-- b1)) * A),1)),B2)) implies (lmlt ((a * (Line (((LineVec2Mx (v |-- b1)) * A),1))),B2)) . k = a * ((lmlt ((Line (((LineVec2Mx (v |-- b1)) * A),1)),B2)) /. k) ) assume A9: k in dom (lmlt ((Line (((LineVec2Mx (v |-- b1)) * A),1)),B2)) ; ::_thesis: (lmlt ((a * (Line (((LineVec2Mx (v |-- b1)) * A),1))),B2)) . k = a * ((lmlt ((Line (((LineVec2Mx (v |-- b1)) * A),1)),B2)) /. k) A10: (lmlt ((Line (((LineVec2Mx (v |-- b1)) * A),1)),B2)) . k = (lmlt ((Line (((LineVec2Mx (v |-- b1)) * A),1)),B2)) /. k by A9, PARTFUN1:def_6; k in dom (Line (((LineVec2Mx (v |-- b1)) * A),1)) by A4, A9, XBOOLE_0:def_4; then A11: (Line (((LineVec2Mx (v |-- b1)) * A),1)) . k = (Line (((LineVec2Mx (v |-- b1)) * A),1)) /. k by PARTFUN1:def_6; k in dom B2 by A4, A9, XBOOLE_0:def_4; then A12: B2 . k = B2 /. k by PARTFUN1:def_6; k in dom (a * (Line (((LineVec2Mx (v |-- b1)) * A),1))) by A5, A4, A9, XBOOLE_0:def_4; then (a * (Line (((LineVec2Mx (v |-- b1)) * A),1))) . k = a * ((Line (((LineVec2Mx (v |-- b1)) * A),1)) /. k) by A11, FVSUM_1:50; hence (lmlt ((a * (Line (((LineVec2Mx (v |-- b1)) * A),1))),B2)) . k = (a * ((Line (((LineVec2Mx (v |-- b1)) * A),1)) /. k)) * (B2 /. k) by A6, A5, A4, A9, A12, FUNCOP_1:22 .= a * (((Line (((LineVec2Mx (v |-- b1)) * A),1)) /. k) * (B2 /. k)) by VECTSP_1:def_16 .= a * ((lmlt ((Line (((LineVec2Mx (v |-- b1)) * A),1)),B2)) /. k) by A9, A11, A12, A10, FUNCOP_1:22 ; ::_thesis: verum end; thus (Mx2Tran ((a * A),b1,B2)) . x = Sum (lmlt ((Line (((LineVec2Mx (v |-- b1)) * (a * A)),1)),B2)) by Def3 .= Sum (lmlt ((Line ((a * ((LineVec2Mx (v |-- b1)) * A)),1)),B2)) by A1, A2, MATRIXR1:22 .= Sum (lmlt ((a * (Line (((LineVec2Mx (v |-- b1)) * A),1))),B2)) by A3, MATRIXR1:20 .= a * (Sum (lmlt ((Line (((LineVec2Mx (v |-- b1)) * A),1)),B2))) by A7, A8, RLVECT_2:67 .= a * ((Mx2Tran (A,b1,B2)) . v) by Def3 .= (a * (Mx2Tran (A,b1,B2))) . x by MATRLIN:def_4 ; ::_thesis: verum end; hence a * (Mx2Tran (A,b1,B2)) = Mx2Tran ((a * A),b1,B2) by FUNCT_2:12; ::_thesis: verum end; theorem :: MATRLIN2:39 for K being Field for V1, V2 being finite-dimensional VectSp of K for b1 being OrdBasis of V1 for b2 being OrdBasis of V2 for A, B being Matrix of len b1, len b2,K st Mx2Tran (A,b1,b2) = Mx2Tran (B,b1,b2) holds A = B proof let K be Field; ::_thesis: for V1, V2 being finite-dimensional VectSp of K for b1 being OrdBasis of V1 for b2 being OrdBasis of V2 for A, B being Matrix of len b1, len b2,K st Mx2Tran (A,b1,b2) = Mx2Tran (B,b1,b2) holds A = B let V1, V2 be finite-dimensional VectSp of K; ::_thesis: for b1 being OrdBasis of V1 for b2 being OrdBasis of V2 for A, B being Matrix of len b1, len b2,K st Mx2Tran (A,b1,b2) = Mx2Tran (B,b1,b2) holds A = B let b1 be OrdBasis of V1; ::_thesis: for b2 being OrdBasis of V2 for A, B being Matrix of len b1, len b2,K st Mx2Tran (A,b1,b2) = Mx2Tran (B,b1,b2) holds A = B let b2 be OrdBasis of V2; ::_thesis: for A, B being Matrix of len b1, len b2,K st Mx2Tran (A,b1,b2) = Mx2Tran (B,b1,b2) holds A = B let A, B be Matrix of len b1, len b2,K; ::_thesis: ( Mx2Tran (A,b1,b2) = Mx2Tran (B,b1,b2) implies A = B ) assume Mx2Tran (A,b1,b2) = Mx2Tran (B,b1,b2) ; ::_thesis: A = B hence A = AutMt ((Mx2Tran (B,b1,b2)),b1,b2) by Th36 .= B by Th36 ; ::_thesis: verum end; theorem :: MATRLIN2:40 for K being Field for V1, V2, V3 being finite-dimensional VectSp of K for b1 being OrdBasis of V1 for b2 being OrdBasis of V2 for B3 being FinSequence of V3 for A being Matrix of len b1, len b2,K for B being Matrix of len b2, len B3,K st width A = len B holds for AB being Matrix of len b1, len B3,K st AB = A * B holds Mx2Tran (AB,b1,B3) = (Mx2Tran (B,b2,B3)) * (Mx2Tran (A,b1,b2)) proof let K be Field; ::_thesis: for V1, V2, V3 being finite-dimensional VectSp of K for b1 being OrdBasis of V1 for b2 being OrdBasis of V2 for B3 being FinSequence of V3 for A being Matrix of len b1, len b2,K for B being Matrix of len b2, len B3,K st width A = len B holds for AB being Matrix of len b1, len B3,K st AB = A * B holds Mx2Tran (AB,b1,B3) = (Mx2Tran (B,b2,B3)) * (Mx2Tran (A,b1,b2)) let V1, V2, V3 be finite-dimensional VectSp of K; ::_thesis: for b1 being OrdBasis of V1 for b2 being OrdBasis of V2 for B3 being FinSequence of V3 for A being Matrix of len b1, len b2,K for B being Matrix of len b2, len B3,K st width A = len B holds for AB being Matrix of len b1, len B3,K st AB = A * B holds Mx2Tran (AB,b1,B3) = (Mx2Tran (B,b2,B3)) * (Mx2Tran (A,b1,b2)) let b1 be OrdBasis of V1; ::_thesis: for b2 being OrdBasis of V2 for B3 being FinSequence of V3 for A being Matrix of len b1, len b2,K for B being Matrix of len b2, len B3,K st width A = len B holds for AB being Matrix of len b1, len B3,K st AB = A * B holds Mx2Tran (AB,b1,B3) = (Mx2Tran (B,b2,B3)) * (Mx2Tran (A,b1,b2)) let b2 be OrdBasis of V2; ::_thesis: for B3 being FinSequence of V3 for A being Matrix of len b1, len b2,K for B being Matrix of len b2, len B3,K st width A = len B holds for AB being Matrix of len b1, len B3,K st AB = A * B holds Mx2Tran (AB,b1,B3) = (Mx2Tran (B,b2,B3)) * (Mx2Tran (A,b1,b2)) let B3 be FinSequence of V3; ::_thesis: for A being Matrix of len b1, len b2,K for B being Matrix of len b2, len B3,K st width A = len B holds for AB being Matrix of len b1, len B3,K st AB = A * B holds Mx2Tran (AB,b1,B3) = (Mx2Tran (B,b2,B3)) * (Mx2Tran (A,b1,b2)) let A be Matrix of len b1, len b2,K; ::_thesis: for B being Matrix of len b2, len B3,K st width A = len B holds for AB being Matrix of len b1, len B3,K st AB = A * B holds Mx2Tran (AB,b1,B3) = (Mx2Tran (B,b2,B3)) * (Mx2Tran (A,b1,b2)) let B be Matrix of len b2, len B3,K; ::_thesis: ( width A = len B implies for AB being Matrix of len b1, len B3,K st AB = A * B holds Mx2Tran (AB,b1,B3) = (Mx2Tran (B,b2,B3)) * (Mx2Tran (A,b1,b2)) ) assume A1: width A = len B ; ::_thesis: for AB being Matrix of len b1, len B3,K st AB = A * B holds Mx2Tran (AB,b1,B3) = (Mx2Tran (B,b2,B3)) * (Mx2Tran (A,b1,b2)) set MB = Mx2Tran (B,b2,B3); set MA = Mx2Tran (A,b1,b2); let AB be Matrix of len b1, len B3,K; ::_thesis: ( AB = A * B implies Mx2Tran (AB,b1,B3) = (Mx2Tran (B,b2,B3)) * (Mx2Tran (A,b1,b2)) ) assume A2: AB = A * B ; ::_thesis: Mx2Tran (AB,b1,B3) = (Mx2Tran (B,b2,B3)) * (Mx2Tran (A,b1,b2)) set MAB = Mx2Tran (AB,b1,B3); now__::_thesis:_for_x_being_set_st_x_in_the_carrier_of_V1_holds_ ((Mx2Tran_(B,b2,B3))_*_(Mx2Tran_(A,b1,b2)))_._x_=_(Mx2Tran_(AB,b1,B3))_._x let x be set ; ::_thesis: ( x in the carrier of V1 implies ((Mx2Tran (B,b2,B3)) * (Mx2Tran (A,b1,b2))) . x = (Mx2Tran (AB,b1,B3)) . x ) assume x in the carrier of V1 ; ::_thesis: ((Mx2Tran (B,b2,B3)) * (Mx2Tran (A,b1,b2))) . x = (Mx2Tran (AB,b1,B3)) . x then reconsider v = x as Element of V1 ; set L = LineVec2Mx (v |-- b1); A3: len A = len b1 by MATRIX_1:def_2; A4: ( width (LineVec2Mx (v |-- b1)) = len (v |-- b1) & len (v |-- b1) = len b1 ) by MATRIX_1:23, MATRLIN:def_7; then ( len (LineVec2Mx (v |-- b1)) = 1 & len ((LineVec2Mx (v |-- b1)) * A) = len (LineVec2Mx (v |-- b1)) ) by A3, MATRIX_1:23, MATRIX_3:def_4; then A5: dom ((LineVec2Mx (v |-- b1)) * A) = Seg 1 by FINSEQ_1:def_3; A6: width ((LineVec2Mx (v |-- b1)) * A) = width A by A4, A3, MATRIX_3:def_4; then A7: ( len B = len b2 & len (Line (((LineVec2Mx (v |-- b1)) * A),1)) = width A ) by CARD_1:def_7, MATRIX_1:def_2; A8: 1 in Seg 1 ; dom ((Mx2Tran (B,b2,B3)) * (Mx2Tran (A,b1,b2))) = the carrier of V1 by FUNCT_2:def_1; hence ((Mx2Tran (B,b2,B3)) * (Mx2Tran (A,b1,b2))) . x = (Mx2Tran (B,b2,B3)) . ((Mx2Tran (A,b1,b2)) . v) by FUNCT_1:12 .= (Mx2Tran (B,b2,B3)) . (Sum (lmlt ((Line (((LineVec2Mx (v |-- b1)) * A),1)),b2))) by Def3 .= Sum (lmlt ((Line (((LineVec2Mx ((Sum (lmlt ((Line (((LineVec2Mx (v |-- b1)) * A),1)),b2))) |-- b2)) * B),1)),B3)) by Def3 .= Sum (lmlt ((Line (((LineVec2Mx (Line (((LineVec2Mx (v |-- b1)) * A),1))) * B),1)),B3)) by A1, A7, MATRLIN:36 .= Sum (lmlt ((Line ((LineVec2Mx (Line ((((LineVec2Mx (v |-- b1)) * A) * B),1))),1)),B3)) by A1, A6, A5, A8, Th35 .= Sum (lmlt ((Line ((LineVec2Mx (Line (((LineVec2Mx (v |-- b1)) * AB),1))),1)),B3)) by A1, A2, A4, A3, MATRIX_3:33 .= Sum (lmlt ((Line (((LineVec2Mx (v |-- b1)) * AB),1)),B3)) by MATRIX15:25 .= (Mx2Tran (AB,b1,B3)) . x by Def3 ; ::_thesis: verum end; hence Mx2Tran (AB,b1,B3) = (Mx2Tran (B,b2,B3)) * (Mx2Tran (A,b1,b2)) by FUNCT_2:12; ::_thesis: verum end; theorem Th41: :: MATRLIN2:41 for K being Field for V2, V1 being finite-dimensional VectSp of K for b1 being OrdBasis of V1 for b2 being OrdBasis of V2 for v1 being Element of V1 for A being Matrix of len b1, len b2,K st len b1 > 0 & len b2 > 0 holds ( v1 in ker (Mx2Tran (A,b1,b2)) iff v1 |-- b1 in Space_of_Solutions_of (A @) ) proof let K be Field; ::_thesis: for V2, V1 being finite-dimensional VectSp of K for b1 being OrdBasis of V1 for b2 being OrdBasis of V2 for v1 being Element of V1 for A being Matrix of len b1, len b2,K st len b1 > 0 & len b2 > 0 holds ( v1 in ker (Mx2Tran (A,b1,b2)) iff v1 |-- b1 in Space_of_Solutions_of (A @) ) let V2, V1 be finite-dimensional VectSp of K; ::_thesis: for b1 being OrdBasis of V1 for b2 being OrdBasis of V2 for v1 being Element of V1 for A being Matrix of len b1, len b2,K st len b1 > 0 & len b2 > 0 holds ( v1 in ker (Mx2Tran (A,b1,b2)) iff v1 |-- b1 in Space_of_Solutions_of (A @) ) let b1 be OrdBasis of V1; ::_thesis: for b2 being OrdBasis of V2 for v1 being Element of V1 for A being Matrix of len b1, len b2,K st len b1 > 0 & len b2 > 0 holds ( v1 in ker (Mx2Tran (A,b1,b2)) iff v1 |-- b1 in Space_of_Solutions_of (A @) ) let b2 be OrdBasis of V2; ::_thesis: for v1 being Element of V1 for A being Matrix of len b1, len b2,K st len b1 > 0 & len b2 > 0 holds ( v1 in ker (Mx2Tran (A,b1,b2)) iff v1 |-- b1 in Space_of_Solutions_of (A @) ) let v1 be Element of V1; ::_thesis: for A being Matrix of len b1, len b2,K st len b1 > 0 & len b2 > 0 holds ( v1 in ker (Mx2Tran (A,b1,b2)) iff v1 |-- b1 in Space_of_Solutions_of (A @) ) let A be Matrix of len b1, len b2,K; ::_thesis: ( len b1 > 0 & len b2 > 0 implies ( v1 in ker (Mx2Tran (A,b1,b2)) iff v1 |-- b1 in Space_of_Solutions_of (A @) ) ) assume that A1: len b1 > 0 and A2: len b2 > 0 ; ::_thesis: ( v1 in ker (Mx2Tran (A,b1,b2)) iff v1 |-- b1 in Space_of_Solutions_of (A @) ) set AT = A @ ; A3: width A = len b2 by A1, MATRIX_1:23; then A4: len (A @) = len b2 by A2, MATRIX_2:10; set L = LineVec2Mx (v1 |-- b1); set M = Mx2Tran (A,b1,b2); set SA = Space_of_Solutions_of (A @); A5: width (LineVec2Mx (v1 |-- b1)) = len (v1 |-- b1) by MATRIX_1:23; A6: len ((len b2) |-> (0. K)) = len b2 by CARD_1:def_7; then A7: width (LineVec2Mx ((len b2) |-> (0. K))) = len b2 by MATRIX_1:23; A8: width (ColVec2Mx ((len b2) |-> (0. K))) = 1 by A2, A6, MATRIX_1:23; A9: len (v1 |-- b1) = len b1 by MATRLIN:def_7; then A10: ( len (ColVec2Mx (v1 |-- b1)) = len (v1 |-- b1) & width (ColVec2Mx (v1 |-- b1)) = 1 ) by A1, MATRIX_1:23; A11: len A = len b1 by A1, MATRIX_1:23; then A12: ( width (A @) = 0 implies len (A @) = 0 ) by A1, A2, A3, MATRIX_2:10; A13: width (A @) = len b1 by A2, A11, A3, MATRIX_2:10; thus ( v1 in ker (Mx2Tran (A,b1,b2)) implies v1 |-- b1 in Space_of_Solutions_of (A @) ) ::_thesis: ( v1 |-- b1 in Space_of_Solutions_of (A @) implies v1 in ker (Mx2Tran (A,b1,b2)) ) proof assume v1 in ker (Mx2Tran (A,b1,b2)) ; ::_thesis: v1 |-- b1 in Space_of_Solutions_of (A @) then (Mx2Tran (A,b1,b2)) . v1 = 0. V2 by RANKNULL:10; then (LineVec2Mx (v1 |-- b1)) * A = LineVec2Mx ((0. V2) |-- b2) by A1, Th32 .= LineVec2Mx ((len b2) |-> (0. K)) by Th20 ; then ColVec2Mx ((len b2) |-> (0. K)) = (A @) * (ColVec2Mx (v1 |-- b1)) by A2, A11, A3, A5, A9, MATRIX_3:22; then ColVec2Mx (v1 |-- b1) in Solutions_of ((A @),(ColVec2Mx ((len b2) |-> (0. K)))) by A13, A9, A10, A8; then v1 |-- b1 in Solutions_of ((A @),((len b2) |-> (0. K))) ; then v1 |-- b1 in the carrier of (Space_of_Solutions_of (A @)) by A4, A12, MATRIX15:def_5; hence v1 |-- b1 in Space_of_Solutions_of (A @) by STRUCT_0:def_5; ::_thesis: verum end; assume v1 |-- b1 in Space_of_Solutions_of (A @) ; ::_thesis: v1 in ker (Mx2Tran (A,b1,b2)) then v1 |-- b1 in the carrier of (Space_of_Solutions_of (A @)) by STRUCT_0:def_5; then v1 |-- b1 in Solutions_of ((A @),((len b2) |-> (0. K))) by A4, A12, MATRIX15:def_5; then ex f being FinSequence of K st ( f = v1 |-- b1 & ColVec2Mx f in Solutions_of ((A @),(ColVec2Mx ((len b2) |-> (0. K)))) ) ; then ex X being Matrix of K st ( X = ColVec2Mx (v1 |-- b1) & len X = width (A @) & width X = width (ColVec2Mx ((len b2) |-> (0. K))) & ColVec2Mx ((len b2) |-> (0. K)) = (A @) * X ) ; then A14: ColVec2Mx ((len b2) |-> (0. K)) = ((LineVec2Mx (v1 |-- b1)) * A) @ by A2, A11, A3, A5, A9, MATRIX_3:22; width ((LineVec2Mx (v1 |-- b1)) * A) = width A by A11, A5, A9, MATRIX_3:def_4; then (LineVec2Mx (v1 |-- b1)) * A = LineVec2Mx ((len b2) |-> (0. K)) by A2, A3, A7, A14, MATRIX_2:12 .= LineVec2Mx ((0. V2) |-- b2) by Th20 ; then LineVec2Mx ((0. V2) |-- b2) = LineVec2Mx (((Mx2Tran (A,b1,b2)) . v1) |-- b2) by A1, Th32; then (0. V2) |-- b2 = Line ((LineVec2Mx (((Mx2Tran (A,b1,b2)) . v1) |-- b2)),1) by MATRIX15:25 .= ((Mx2Tran (A,b1,b2)) . v1) |-- b2 by MATRIX15:25 ; then (Mx2Tran (A,b1,b2)) . v1 = 0. V2 by MATRLIN:34; hence v1 in ker (Mx2Tran (A,b1,b2)) by RANKNULL:10; ::_thesis: verum end; theorem Th42: :: MATRLIN2:42 for K being Field for V1 being finite-dimensional VectSp of K holds ( V1 is trivial iff dim V1 = 0 ) proof let K be Field; ::_thesis: for V1 being finite-dimensional VectSp of K holds ( V1 is trivial iff dim V1 = 0 ) let V1 be finite-dimensional VectSp of K; ::_thesis: ( V1 is trivial iff dim V1 = 0 ) hereby ::_thesis: ( dim V1 = 0 implies V1 is trivial ) assume A1: V1 is trivial ; ::_thesis: dim V1 = 0 the carrier of V1 c= {(0. V1)} proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in the carrier of V1 or x in {(0. V1)} ) assume A2: x in the carrier of V1 ; ::_thesis: x in {(0. V1)} x = 0. V1 by A1, A2, STRUCT_0:def_18; hence x in {(0. V1)} by TARSKI:def_1; ::_thesis: verum end; then the carrier of ((Omega). V1) = {(0. V1)} by ZFMISC_1:33 .= the carrier of ((0). V1) by VECTSP_4:def_3 ; then (Omega). V1 = (0). V1 by VECTSP_4:29; hence dim V1 = 0 by VECTSP_9:29; ::_thesis: verum end; assume dim V1 = 0 ; ::_thesis: V1 is trivial then A3: (Omega). V1 = (0). V1 by VECTSP_9:29; now__::_thesis:_for_v1_being_Element_of_V1_holds_v1_=_0._V1 let v1 be Element of V1; ::_thesis: v1 = 0. V1 v1 in (0). V1 by A3, STRUCT_0:def_5; hence v1 = 0. V1 by VECTSP_4:35; ::_thesis: verum end; hence V1 is trivial by STRUCT_0:def_18; ::_thesis: verum end; theorem Th43: :: MATRLIN2:43 for K being Field for V1, V2 being VectSp of K for f being linear-transformation of V1,V2 holds ( f is one-to-one iff ker f = (0). V1 ) proof let K be Field; ::_thesis: for V1, V2 being VectSp of K for f being linear-transformation of V1,V2 holds ( f is one-to-one iff ker f = (0). V1 ) let V1, V2 be VectSp of K; ::_thesis: for f being linear-transformation of V1,V2 holds ( f is one-to-one iff ker f = (0). V1 ) let f be linear-transformation of V1,V2; ::_thesis: ( f is one-to-one iff ker f = (0). V1 ) ( ker f = (0). V1 implies f is one-to-one ) proof assume A1: ker f = (0). V1 ; ::_thesis: f is one-to-one let x be set ; :: according to FUNCT_1:def_4 ::_thesis: for b1 being set holds ( not x in dom f or not b1 in dom f or not f . x = f . b1 or x = b1 ) let y be set ; ::_thesis: ( not x in dom f or not y in dom f or not f . x = f . y or x = y ) assume that A2: ( x in dom f & y in dom f ) and A3: f . x = f . y ; ::_thesis: x = y reconsider x9 = x, y9 = y as Element of V1 by A2, FUNCT_2:def_1; x9 - y9 in ker f by A3, RANKNULL:17; then x9 - y9 in the carrier of ((0). V1) by A1, STRUCT_0:def_5; then x9 - y9 in {(0. V1)} by VECTSP_4:def_3; then x9 + (- y9) = 0. V1 by TARSKI:def_1; hence x = - (- y9) by VECTSP_1:16 .= y by RLVECT_1:17 ; ::_thesis: verum end; hence ( f is one-to-one iff ker f = (0). V1 ) by RANKNULL:15; ::_thesis: verum end; registration let K be Field; let V1, V2 be VectSp of K; let f, g be linear-transformation of V1,V2; clusterf + g -> additive homogeneous ; coherence ( f + g is homogeneous & f + g is additive ) proof A1: now__::_thesis:_for_a_being_Scalar_of_K for_v1_being_Vector_of_V1_holds_(f_+_g)_._(a_*_v1)_=_a_*_((f_+_g)_._v1) let a be Scalar of K; ::_thesis: for v1 being Vector of V1 holds (f + g) . (a * v1) = a * ((f + g) . v1) let v1 be Vector of V1; ::_thesis: (f + g) . (a * v1) = a * ((f + g) . v1) thus (f + g) . (a * v1) = (f . (a * v1)) + (g . (a * v1)) by MATRLIN:def_3 .= (a * (f . v1)) + (g . (a * v1)) by MOD_2:def_2 .= (a * (f . v1)) + (a * (g . v1)) by MOD_2:def_2 .= a * ((f . v1) + (g . v1)) by VECTSP_1:def_14 .= a * ((f + g) . v1) by MATRLIN:def_3 ; ::_thesis: verum end; now__::_thesis:_for_v1,_w1_being_Vector_of_V1_holds_(f_+_g)_._(v1_+_w1)_=_((f_+_g)_._v1)_+_((f_+_g)_._w1) let v1, w1 be Vector of V1; ::_thesis: (f + g) . (v1 + w1) = ((f + g) . v1) + ((f + g) . w1) thus (f + g) . (v1 + w1) = (f . (v1 + w1)) + (g . (v1 + w1)) by MATRLIN:def_3 .= ((f . v1) + (f . w1)) + (g . (v1 + w1)) by VECTSP_1:def_20 .= ((f . v1) + (f . w1)) + ((g . v1) + (g . w1)) by VECTSP_1:def_20 .= (f . v1) + ((f . w1) + ((g . v1) + (g . w1))) by RLVECT_1:def_3 .= (f . v1) + ((g . v1) + ((g . w1) + (f . w1))) by RLVECT_1:def_3 .= ((f . v1) + (g . v1)) + ((f . w1) + (g . w1)) by RLVECT_1:def_3 .= ((f + g) . v1) + ((f . w1) + (g . w1)) by MATRLIN:def_3 .= ((f + g) . v1) + ((f + g) . w1) by MATRLIN:def_3 ; ::_thesis: verum end; then ( f + g is additive & f + g is homogeneous ) by A1, VECTSP_1:def_20, MOD_2:def_2; hence ( f + g is homogeneous & f + g is additive ) ; ::_thesis: verum end; end; registration let K be Field; let V1, V2 be VectSp of K; let f be linear-transformation of V1,V2; let a be Element of K; clustera * f -> additive homogeneous ; coherence ( a * f is homogeneous & a * f is additive ) proof A1: now__::_thesis:_for_b_being_Scalar_of_K for_v1_being_Vector_of_V1_holds_(a_*_f)_._(b_*_v1)_=_b_*_((a_*_f)_._v1) let b be Scalar of K; ::_thesis: for v1 being Vector of V1 holds (a * f) . (b * v1) = b * ((a * f) . v1) let v1 be Vector of V1; ::_thesis: (a * f) . (b * v1) = b * ((a * f) . v1) thus (a * f) . (b * v1) = a * (f . (b * v1)) by MATRLIN:def_4 .= a * (b * (f . v1)) by MOD_2:def_2 .= (a * b) * (f . v1) by VECTSP_1:def_16 .= b * (a * (f . v1)) by VECTSP_1:def_16 .= b * ((a * f) . v1) by MATRLIN:def_4 ; ::_thesis: verum end; now__::_thesis:_for_v1,_w1_being_Vector_of_V1_holds_(a_*_f)_._(v1_+_w1)_=_((a_*_f)_._v1)_+_((a_*_f)_._w1) let v1, w1 be Vector of V1; ::_thesis: (a * f) . (v1 + w1) = ((a * f) . v1) + ((a * f) . w1) thus (a * f) . (v1 + w1) = a * (f . (v1 + w1)) by MATRLIN:def_4 .= a * ((f . v1) + (f . w1)) by VECTSP_1:def_20 .= (a * (f . v1)) + (a * (f . w1)) by VECTSP_1:def_14 .= ((a * f) . v1) + (a * (f . w1)) by MATRLIN:def_4 .= ((a * f) . v1) + ((a * f) . w1) by MATRLIN:def_4 ; ::_thesis: verum end; then ( a * f is additive & a * f is homogeneous ) by A1, VECTSP_1:def_20, MOD_2:def_2; hence ( a * f is homogeneous & a * f is additive ) ; ::_thesis: verum end; end; definition let K be Field; let V1, V2, V3 be VectSp of K; let f1 be linear-transformation of V1,V2; let f2 be linear-transformation of V2,V3; :: original: (#) redefine funcf2 * f1 -> linear-transformation of V1,V3; coherence f1 (#) f2 is linear-transformation of V1,V3 by MOD_2:2; end; theorem Th44: :: MATRLIN2:44 for K being Field for V1, V2 being finite-dimensional VectSp of K for b1 being OrdBasis of V1 for b2 being OrdBasis of V2 for A being Matrix of len b1, len b2,K st the_rank_of A = len b1 holds Mx2Tran (A,b1,b2) is one-to-one proof let K be Field; ::_thesis: for V1, V2 being finite-dimensional VectSp of K for b1 being OrdBasis of V1 for b2 being OrdBasis of V2 for A being Matrix of len b1, len b2,K st the_rank_of A = len b1 holds Mx2Tran (A,b1,b2) is one-to-one let V1, V2 be finite-dimensional VectSp of K; ::_thesis: for b1 being OrdBasis of V1 for b2 being OrdBasis of V2 for A being Matrix of len b1, len b2,K st the_rank_of A = len b1 holds Mx2Tran (A,b1,b2) is one-to-one let b1 be OrdBasis of V1; ::_thesis: for b2 being OrdBasis of V2 for A being Matrix of len b1, len b2,K st the_rank_of A = len b1 holds Mx2Tran (A,b1,b2) is one-to-one let b2 be OrdBasis of V2; ::_thesis: for A being Matrix of len b1, len b2,K st the_rank_of A = len b1 holds Mx2Tran (A,b1,b2) is one-to-one let A be Matrix of len b1, len b2,K; ::_thesis: ( the_rank_of A = len b1 implies Mx2Tran (A,b1,b2) is one-to-one ) assume A1: the_rank_of A = len b1 ; ::_thesis: Mx2Tran (A,b1,b2) is one-to-one set S = Space_of_Solutions_of (A @); set M = Mx2Tran (A,b1,b2); A2: now__::_thesis:_the_carrier_of_(ker_(Mx2Tran_(A,b1,b2)))_c=_{(0._V1)} percases ( len b1 = 0 or len b1 > 0 ) ; suppose len b1 = 0 ; ::_thesis: the carrier of (ker (Mx2Tran (A,b1,b2))) c= {(0. V1)} then dim V1 = 0 by Th21; then A3: (Omega). V1 = (0). V1 by VECTSP_9:29; the carrier of (ker (Mx2Tran (A,b1,b2))) c= the carrier of V1 by VECTSP_4:def_2; hence the carrier of (ker (Mx2Tran (A,b1,b2))) c= {(0. V1)} by A3, VECTSP_4:def_3; ::_thesis: verum end; supposeA4: len b1 > 0 ; ::_thesis: the carrier of (ker (Mx2Tran (A,b1,b2))) c= {(0. V1)} A5: len b1 <= width A by A1, MATRIX13:74; then A6: width (A @) = len A by A4, MATRIX_2:10; A7: len A = len b1 by A4, MATRIX_1:23; A8: width A = len b2 by A4, MATRIX_1:23; thus the carrier of (ker (Mx2Tran (A,b1,b2))) c= {(0. V1)} ::_thesis: verum proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in the carrier of (ker (Mx2Tran (A,b1,b2))) or x in {(0. V1)} ) assume A9: x in the carrier of (ker (Mx2Tran (A,b1,b2))) ; ::_thesis: x in {(0. V1)} the carrier of (ker (Mx2Tran (A,b1,b2))) c= the carrier of V1 by VECTSP_4:def_2; then reconsider v = x as Element of V1 by A9; dim (Space_of_Solutions_of (A @)) = (len b1) - (the_rank_of (A @)) by A4, A7, A6, MATRIX15:68 .= (len b1) - (len b1) by A1, MATRIX13:84 .= 0 ; then A10: (Omega). (Space_of_Solutions_of (A @)) = (0). (Space_of_Solutions_of (A @)) by VECTSP_9:29; v in ker (Mx2Tran (A,b1,b2)) by A9, STRUCT_0:def_5; then v |-- b1 in Space_of_Solutions_of (A @) by A4, A8, A5, Th41; then v |-- b1 in the carrier of ((0). (Space_of_Solutions_of (A @))) by A10, STRUCT_0:def_5; then v |-- b1 in the carrier of ((0). ((len b1) -VectSp_over K)) by A7, A6, VECTSP_4:36; then v |-- b1 in {(0. ((len b1) -VectSp_over K))} by VECTSP_4:def_3; then v |-- b1 = 0. ((len b1) -VectSp_over K) by TARSKI:def_1 .= (len b1) |-> (0. K) by MATRIX13:102 .= (0. V1) |-- b1 by Th20 ; then v = 0. V1 by MATRLIN:34; hence x in {(0. V1)} by TARSKI:def_1; ::_thesis: verum end; end; end; end; 0. V1 in ker (Mx2Tran (A,b1,b2)) by RANKNULL:11; then 0. V1 in the carrier of (ker (Mx2Tran (A,b1,b2))) by STRUCT_0:def_5; then {(0. V1)} c= the carrier of (ker (Mx2Tran (A,b1,b2))) by ZFMISC_1:31; then the carrier of (ker (Mx2Tran (A,b1,b2))) = {(0. V1)} by A2, XBOOLE_0:def_10 .= the carrier of ((0). V1) by VECTSP_4:def_3 ; then ker (Mx2Tran (A,b1,b2)) = (0). V1 by VECTSP_4:29; hence Mx2Tran (A,b1,b2) is one-to-one by Th43; ::_thesis: verum end; Lm6: for n being Nat for K being Field holds the_rank_of (1. (K,n)) = n proof let n be Nat; ::_thesis: for K being Field holds the_rank_of (1. (K,n)) = n let K be Field; ::_thesis: the_rank_of (1. (K,n)) = n A1: 1_ K <> 0. K ; ( n + 0 > 0 or n = 0 ) ; then A2: ( n >= 1 or n = 0 ) by NAT_1:19; n in NAT by ORDINAL1:def_12; then Det (1. (K,n)) = 1_ K by A2, MATRIXR2:41, MATRIX_7:16; hence the_rank_of (1. (K,n)) = n by A1, MATRIX13:83; ::_thesis: verum end; theorem Th45: :: MATRLIN2:45 for n being Nat for K being Field holds MX2FinS (1. (K,n)) is OrdBasis of n -VectSp_over K proof let n be Nat; ::_thesis: for K being Field holds MX2FinS (1. (K,n)) is OrdBasis of n -VectSp_over K let K be Field; ::_thesis: MX2FinS (1. (K,n)) is OrdBasis of n -VectSp_over K set ONE = 1. (K,n); A1: the_rank_of (1. (K,n)) = n by Lm6; then A2: 1. (K,n) is one-to-one by MATRIX13:105; for i, j being Nat st [i,j] in Indices (1. (K,n)) & (1. (K,n)) * (i,j) <> 0. K holds i = j by MATRIX_1:def_11; then 1. (K,n) is diagonal by MATRIX_1:def_14; then lines (1. (K,n)) is Basis of n -VectSp_over K by A1, MATRIX13:111; hence MX2FinS (1. (K,n)) is OrdBasis of n -VectSp_over K by A2, MATRLIN:def_2; ::_thesis: verum end; theorem Th46: :: MATRLIN2:46 for K being Field for V2 being finite-dimensional VectSp of K for b2 being OrdBasis of V2 for M being OrdBasis of (len b2) -VectSp_over K st M = MX2FinS (1. (K,(len b2))) holds for v1 being Vector of ((len b2) -VectSp_over K) holds v1 |-- M = v1 proof let K be Field; ::_thesis: for V2 being finite-dimensional VectSp of K for b2 being OrdBasis of V2 for M being OrdBasis of (len b2) -VectSp_over K st M = MX2FinS (1. (K,(len b2))) holds for v1 being Vector of ((len b2) -VectSp_over K) holds v1 |-- M = v1 let V2 be finite-dimensional VectSp of K; ::_thesis: for b2 being OrdBasis of V2 for M being OrdBasis of (len b2) -VectSp_over K st M = MX2FinS (1. (K,(len b2))) holds for v1 being Vector of ((len b2) -VectSp_over K) holds v1 |-- M = v1 let b2 be OrdBasis of V2; ::_thesis: for M being OrdBasis of (len b2) -VectSp_over K st M = MX2FinS (1. (K,(len b2))) holds for v1 being Vector of ((len b2) -VectSp_over K) holds v1 |-- M = v1 let M be OrdBasis of (len b2) -VectSp_over K; ::_thesis: ( M = MX2FinS (1. (K,(len b2))) implies for v1 being Vector of ((len b2) -VectSp_over K) holds v1 |-- M = v1 ) assume A1: M = MX2FinS (1. (K,(len b2))) ; ::_thesis: for v1 being Vector of ((len b2) -VectSp_over K) holds v1 |-- M = v1 let v1 be Vector of ((len b2) -VectSp_over K); ::_thesis: v1 |-- M = v1 set vM = v1 |-- M; consider KL being Linear_Combination of (len b2) -VectSp_over K such that A2: ( v1 = Sum KL & Carrier KL c= rng M ) and A3: for k being Nat st 1 <= k & k <= len (v1 |-- M) holds (v1 |-- M) /. k = KL . (M /. k) by MATRLIN:def_7; reconsider t1 = v1 as Element of (len b2) -tuples_on the carrier of K by MATRIX13:102; A4: len t1 = len b2 by CARD_1:def_7; A5: ( len M = dim ((len b2) -VectSp_over K) & dim ((len b2) -VectSp_over K) = len b2 ) by Th21, MATRIX13:112; A6: len (v1 |-- M) = len M by MATRLIN:def_7; now__::_thesis:_for_i_being_Nat_st_1_<=_i_&_i_<=_len_t1_holds_ t1_._i_=_(v1_|--_M)_._i A7: dom M = dom (v1 |-- M) by A6, FINSEQ_3:29; A8: the_rank_of (1. (K,(len b2))) = len b2 by Lm6; set F = FinS2MX (KL (#) M); A9: Indices (1. (K,(len b2))) = [:(Seg (len b2)),(Seg (len b2)):] by MATRIX_1:24; let i be Nat; ::_thesis: ( 1 <= i & i <= len t1 implies t1 . i = (v1 |-- M) . i ) assume A10: ( 1 <= i & i <= len t1 ) ; ::_thesis: t1 . i = (v1 |-- M) . i A11: i in Seg (len b2) by A4, A10, FINSEQ_1:1; then A12: [i,i] in [:(Seg (len b2)),(Seg (len b2)):] by ZFMISC_1:87; A13: width (1. (K,(len b2))) = len b2 by MATRIX_1:24; then A14: (Line ((1. (K,(len b2))),i)) . i = (1. (K,(len b2))) * (i,i) by A11, MATRIX_1:def_7 .= 1_ K by A9, A12, MATRIX_1:def_11 ; A15: len (Col ((FinS2MX (KL (#) M)),i)) = len (FinS2MX (KL (#) M)) by CARD_1:def_7; then A16: dom (Col ((FinS2MX (KL (#) M)),i)) = dom (FinS2MX (KL (#) M)) by FINSEQ_3:29; A17: len (FinS2MX (KL (#) M)) = len M by VECTSP_6:def_5; then A18: dom (FinS2MX (KL (#) M)) = dom M by FINSEQ_3:29; A19: i in dom (Col ((FinS2MX (KL (#) M)),i)) by A4, A5, A10, A17, A15, FINSEQ_3:25; A20: width (FinS2MX (KL (#) M)) = len b2 by A5, A17, MATRIX_1:24; now__::_thesis:_for_j_being_Nat_st_j_in_dom_(Col_((FinS2MX_(KL_(#)_M)),i))_&_j_<>_i_holds_ (Col_((FinS2MX_(KL_(#)_M)),i))_._j_=_0._K let j be Nat; ::_thesis: ( j in dom (Col ((FinS2MX (KL (#) M)),i)) & j <> i implies (Col ((FinS2MX (KL (#) M)),i)) . j = 0. K ) assume that A21: j in dom (Col ((FinS2MX (KL (#) M)),i)) and A22: j <> i ; ::_thesis: (Col ((FinS2MX (KL (#) M)),i)) . j = 0. K A23: dom (Col ((FinS2MX (KL (#) M)),i)) = Seg (len b2) by A5, A17, A15, FINSEQ_1:def_3; then A24: [j,i] in [:(Seg (len b2)),(Seg (len b2)):] by A11, A21, ZFMISC_1:87; A25: Line ((FinS2MX (KL (#) M)),j) = (KL (#) M) . j by A5, A17, A21, A23, MATRIX_2:8 .= (KL . (M /. j)) * (M /. j) by A16, A21, VECTSP_6:def_5 ; A26: (Col ((FinS2MX (KL (#) M)),i)) . j = (FinS2MX (KL (#) M)) * (j,i) by A16, A21, MATRIX_1:def_8 .= (Line ((FinS2MX (KL (#) M)),j)) . i by A11, A20, MATRIX_1:def_7 ; A27: (Line ((1. (K,(len b2))),j)) . i = (1. (K,(len b2))) * (j,i) by A11, A13, MATRIX_1:def_7 .= 0. K by A9, A22, A24, MATRIX_1:def_11 ; M /. j = M . j by A16, A18, A21, PARTFUN1:def_6 .= Line ((1. (K,(len b2))),j) by A1, A21, A23, MATRIX_2:8 ; hence (Col ((FinS2MX (KL (#) M)),i)) . j = ((KL . (M /. j)) * (Line ((1. (K,(len b2))),j))) . i by A13, A26, A25, MATRIX13:102 .= (KL . (M /. j)) * (0. K) by A11, A13, A27, FVSUM_1:51 .= 0. K by VECTSP_1:6 ; ::_thesis: verum end; then A28: (Col ((FinS2MX (KL (#) M)),i)) . i = Sum (Col ((FinS2MX (KL (#) M)),i)) by A19, MATRIX_3:12 .= v1 . i by A1, A2, A11, A8, MATRIX13:105, MATRIX13:107 ; A29: Line ((FinS2MX (KL (#) M)),i) = (KL (#) M) . i by A5, A11, A17, MATRIX_2:8 .= (KL . (M /. i)) * (M /. i) by A19, A16, VECTSP_6:def_5 ; A30: (Col ((FinS2MX (KL (#) M)),i)) . i = (FinS2MX (KL (#) M)) * (i,i) by A19, A16, MATRIX_1:def_8 .= (Line ((FinS2MX (KL (#) M)),i)) . i by A11, A20, MATRIX_1:def_7 ; M /. i = M . i by A19, A16, A18, PARTFUN1:def_6 .= Line ((1. (K,(len b2))),i) by A1, A11, MATRIX_2:8 ; then (Col ((FinS2MX (KL (#) M)),i)) . i = ((KL . (M /. i)) * (Line ((1. (K,(len b2))),i))) . i by A30, A13, A29, MATRIX13:102 .= (KL . (M /. i)) * (1_ K) by A11, A13, A14, FVSUM_1:51 .= KL . (M /. i) by VECTSP_1:def_4 ; hence t1 . i = (v1 |-- M) /. i by A3, A4, A6, A5, A10, A28 .= (v1 |-- M) . i by A19, A16, A18, A7, PARTFUN1:def_6 ; ::_thesis: verum end; hence v1 |-- M = v1 by A4, A6, A5, FINSEQ_1:14; ::_thesis: verum end; theorem Th47: :: MATRLIN2:47 for K being Field for V2, V1 being finite-dimensional VectSp of K for f being Function of V1,V2 for b1 being OrdBasis of V1 for b2 being OrdBasis of V2 for v1 being Element of V1 for M being OrdBasis of (len b2) -VectSp_over K st M = MX2FinS (1. (K,(len b2))) holds for A being Matrix of len b1, len M,K st A = AutMt (f,b1,b2) & f is additive & f is homogeneous holds (Mx2Tran (A,b1,M)) . v1 = (f . v1) |-- b2 proof let K be Field; ::_thesis: for V2, V1 being finite-dimensional VectSp of K for f being Function of V1,V2 for b1 being OrdBasis of V1 for b2 being OrdBasis of V2 for v1 being Element of V1 for M being OrdBasis of (len b2) -VectSp_over K st M = MX2FinS (1. (K,(len b2))) holds for A being Matrix of len b1, len M,K st A = AutMt (f,b1,b2) & f is additive & f is homogeneous holds (Mx2Tran (A,b1,M)) . v1 = (f . v1) |-- b2 let V2, V1 be finite-dimensional VectSp of K; ::_thesis: for f being Function of V1,V2 for b1 being OrdBasis of V1 for b2 being OrdBasis of V2 for v1 being Element of V1 for M being OrdBasis of (len b2) -VectSp_over K st M = MX2FinS (1. (K,(len b2))) holds for A being Matrix of len b1, len M,K st A = AutMt (f,b1,b2) & f is additive & f is homogeneous holds (Mx2Tran (A,b1,M)) . v1 = (f . v1) |-- b2 let f be Function of V1,V2; ::_thesis: for b1 being OrdBasis of V1 for b2 being OrdBasis of V2 for v1 being Element of V1 for M being OrdBasis of (len b2) -VectSp_over K st M = MX2FinS (1. (K,(len b2))) holds for A being Matrix of len b1, len M,K st A = AutMt (f,b1,b2) & f is additive & f is homogeneous holds (Mx2Tran (A,b1,M)) . v1 = (f . v1) |-- b2 let b1 be OrdBasis of V1; ::_thesis: for b2 being OrdBasis of V2 for v1 being Element of V1 for M being OrdBasis of (len b2) -VectSp_over K st M = MX2FinS (1. (K,(len b2))) holds for A being Matrix of len b1, len M,K st A = AutMt (f,b1,b2) & f is additive & f is homogeneous holds (Mx2Tran (A,b1,M)) . v1 = (f . v1) |-- b2 let b2 be OrdBasis of V2; ::_thesis: for v1 being Element of V1 for M being OrdBasis of (len b2) -VectSp_over K st M = MX2FinS (1. (K,(len b2))) holds for A being Matrix of len b1, len M,K st A = AutMt (f,b1,b2) & f is additive & f is homogeneous holds (Mx2Tran (A,b1,M)) . v1 = (f . v1) |-- b2 let v1 be Element of V1; ::_thesis: for M being OrdBasis of (len b2) -VectSp_over K st M = MX2FinS (1. (K,(len b2))) holds for A being Matrix of len b1, len M,K st A = AutMt (f,b1,b2) & f is additive & f is homogeneous holds (Mx2Tran (A,b1,M)) . v1 = (f . v1) |-- b2 let M be OrdBasis of (len b2) -VectSp_over K; ::_thesis: ( M = MX2FinS (1. (K,(len b2))) implies for A being Matrix of len b1, len M,K st A = AutMt (f,b1,b2) & f is additive & f is homogeneous holds (Mx2Tran (A,b1,M)) . v1 = (f . v1) |-- b2 ) assume A1: M = MX2FinS (1. (K,(len b2))) ; ::_thesis: for A being Matrix of len b1, len M,K st A = AutMt (f,b1,b2) & f is additive & f is homogeneous holds (Mx2Tran (A,b1,M)) . v1 = (f . v1) |-- b2 let A be Matrix of len b1, len M,K; ::_thesis: ( A = AutMt (f,b1,b2) & f is additive & f is homogeneous implies (Mx2Tran (A,b1,M)) . v1 = (f . v1) |-- b2 ) assume that A2: A = AutMt (f,b1,b2) and A3: ( f is additive & f is homogeneous ) ; ::_thesis: (Mx2Tran (A,b1,M)) . v1 = (f . v1) |-- b2 reconsider f9 = f as linear-transformation of V1,V2 by A3; set MM = Mx2Tran (A,b1,M); percases ( len b1 = 0 or len b1 > 0 ) ; supposeA4: len b1 = 0 ; ::_thesis: (Mx2Tran (A,b1,M)) . v1 = (f . v1) |-- b2 then dim V1 = 0 by Th21; then (Omega). V1 = (0). V1 by VECTSP_9:29; then the carrier of V1 = {(0. V1)} by VECTSP_4:def_3; then v1 = 0. V1 by TARSKI:def_1; then v1 in ker f9 by RANKNULL:11; hence (f . v1) |-- b2 = (0. V2) |-- b2 by RANKNULL:10 .= (len b2) |-> (0. K) by Th20 .= 0. ((len b2) -VectSp_over K) by MATRIX13:102 .= (Mx2Tran (A,b1,M)) . v1 by A4, Th33 ; ::_thesis: verum end; supposeA5: len b1 > 0 ; ::_thesis: (Mx2Tran (A,b1,M)) . v1 = (f . v1) |-- b2 then LineVec2Mx (((Mx2Tran (A,b1,M)) . v1) |-- M) = (LineVec2Mx (v1 |-- b1)) * A by Th32 .= LineVec2Mx ((f . v1) |-- b2) by A2, A3, A5, Th31 ; hence (f . v1) |-- b2 = Line ((LineVec2Mx (((Mx2Tran (A,b1,M)) . v1) |-- M)),1) by MATRIX15:25 .= ((Mx2Tran (A,b1,M)) . v1) |-- M by MATRIX15:25 .= (Mx2Tran (A,b1,M)) . v1 by A1, Th46 ; ::_thesis: verum end; end; end; definition let K be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; let V1, V2 be non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed VectSpStr over K; let W be Subspace of V1; let f be Function of V1,V2; :: original: | redefine funcf | W -> Function of W,V2; coherence f | W is Function of W,V2 proof the carrier of W c= the carrier of V1 by VECTSP_4:def_2; hence f | W is Function of W,V2 by FUNCT_2:32; ::_thesis: verum end; end; definition let K be Field; let V1, V2 be VectSp of K; let W be Subspace of V1; let f be linear-transformation of V1,V2; :: original: | redefine funcf | W -> linear-transformation of W,V2; coherence f | W is linear-transformation of W,V2 proof set F = f | W; A1: dom (f | W) = the carrier of W by FUNCT_2:def_1; A2: now__::_thesis:_for_a_being_Scalar_of_K for_v1_being_Vector_of_W_holds_(f_|_W)_._(a_*_v1)_=_a_*_((f_|_W)_._v1) let a be Scalar of K; ::_thesis: for v1 being Vector of W holds (f | W) . (a * v1) = a * ((f | W) . v1) let v1 be Vector of W; ::_thesis: (f | W) . (a * v1) = a * ((f | W) . v1) reconsider v2 = v1 as Vector of V1 by VECTSP_4:10; a * v1 = a * v2 by VECTSP_4:14; hence (f | W) . (a * v1) = f . (a * v2) by A1, FUNCT_1:47 .= a * (f . v2) by MOD_2:def_2 .= a * ((f | W) . v1) by A1, FUNCT_1:47 ; ::_thesis: verum end; now__::_thesis:_for_v1,_w1_being_Vector_of_W_holds_(f_|_W)_._(v1_+_w1)_=_((f_|_W)_._v1)_+_((f_|_W)_._w1) let v1, w1 be Vector of W; ::_thesis: (f | W) . (v1 + w1) = ((f | W) . v1) + ((f | W) . w1) reconsider v2 = v1, w2 = w1 as Vector of V1 by VECTSP_4:10; v1 + w1 = v2 + w2 by VECTSP_4:13; hence (f | W) . (v1 + w1) = f . (v2 + w2) by A1, FUNCT_1:47 .= (f . v2) + (f . w2) by VECTSP_1:def_20 .= ((f | W) . v1) + (f . w2) by A1, FUNCT_1:47 .= ((f | W) . v1) + ((f | W) . w1) by A1, FUNCT_1:47 ; ::_thesis: verum end; then ( f | W is additive & f | W is homogeneous ) by A2, VECTSP_1:def_20, MOD_2:def_2; hence f | W is linear-transformation of W,V2 ; ::_thesis: verum end; end; Lm7: for K being Field for V1, V2 being finite-dimensional VectSp of K for b1 being OrdBasis of V1 for b2 being OrdBasis of V2 for A being Matrix of len b1, len b2,K st len b1 > 0 & len b2 > 0 holds nullity (Mx2Tran (A,b1,b2)) = (len b1) - (the_rank_of A) proof let K be Field; ::_thesis: for V1, V2 being finite-dimensional VectSp of K for b1 being OrdBasis of V1 for b2 being OrdBasis of V2 for A being Matrix of len b1, len b2,K st len b1 > 0 & len b2 > 0 holds nullity (Mx2Tran (A,b1,b2)) = (len b1) - (the_rank_of A) let V1, V2 be finite-dimensional VectSp of K; ::_thesis: for b1 being OrdBasis of V1 for b2 being OrdBasis of V2 for A being Matrix of len b1, len b2,K st len b1 > 0 & len b2 > 0 holds nullity (Mx2Tran (A,b1,b2)) = (len b1) - (the_rank_of A) let b1 be OrdBasis of V1; ::_thesis: for b2 being OrdBasis of V2 for A being Matrix of len b1, len b2,K st len b1 > 0 & len b2 > 0 holds nullity (Mx2Tran (A,b1,b2)) = (len b1) - (the_rank_of A) let b2 be OrdBasis of V2; ::_thesis: for A being Matrix of len b1, len b2,K st len b1 > 0 & len b2 > 0 holds nullity (Mx2Tran (A,b1,b2)) = (len b1) - (the_rank_of A) set I = id V1; reconsider BB = MX2FinS (1. (K,(len b1))) as OrdBasis of (len b1) -VectSp_over K by Th45; let A be Matrix of len b1, len b2,K; ::_thesis: ( len b1 > 0 & len b2 > 0 implies nullity (Mx2Tran (A,b1,b2)) = (len b1) - (the_rank_of A) ) assume that A1: len b1 > 0 and A2: len b2 > 0 ; ::_thesis: nullity (Mx2Tran (A,b1,b2)) = (len b1) - (the_rank_of A) len BB = dim ((len b1) -VectSp_over K) by Th21 .= len b1 by MATRIX13:112 ; then reconsider AI = AutMt ((id V1),b1,b1) as Matrix of len b1, len BB,K ; A3: ( AutMt ((id V1),b1,b1) = 1. (K,(len b1)) & 0. K <> 1_ K ) by Th28; (len b1) + 0 > 0 by A1; then len b1 >= 1 by NAT_1:19; then Det (1. (K,(len b1))) = 1_ K by MATRIX_7:16; then A4: the_rank_of AI = len b1 by A3, MATRIX13:83; set S = Space_of_Solutions_of (A @); set KER = ker (Mx2Tran (A,b1,b2)); set MAI = Mx2Tran (AI,b1,BB); set MK = (Mx2Tran (AI,b1,BB)) | (ker (Mx2Tran (A,b1,b2))); A5: dom ((Mx2Tran (AI,b1,BB)) | (ker (Mx2Tran (A,b1,b2)))) = the carrier of (ker (Mx2Tran (A,b1,b2))) by FUNCT_2:def_1; A6: the carrier of (im ((Mx2Tran (AI,b1,BB)) | (ker (Mx2Tran (A,b1,b2))))) c= the carrier of (Space_of_Solutions_of (A @)) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in the carrier of (im ((Mx2Tran (AI,b1,BB)) | (ker (Mx2Tran (A,b1,b2))))) or x in the carrier of (Space_of_Solutions_of (A @)) ) assume A7: x in the carrier of (im ((Mx2Tran (AI,b1,BB)) | (ker (Mx2Tran (A,b1,b2))))) ; ::_thesis: x in the carrier of (Space_of_Solutions_of (A @)) the carrier of (im ((Mx2Tran (AI,b1,BB)) | (ker (Mx2Tran (A,b1,b2))))) c= the carrier of ((len b1) -VectSp_over K) by VECTSP_4:def_2; then reconsider v = x as Element of ((len b1) -VectSp_over K) by A7; v in im ((Mx2Tran (AI,b1,BB)) | (ker (Mx2Tran (A,b1,b2)))) by A7, STRUCT_0:def_5; then consider w being Element of (ker (Mx2Tran (A,b1,b2))) such that A8: v = ((Mx2Tran (AI,b1,BB)) | (ker (Mx2Tran (A,b1,b2)))) . w by RANKNULL:13; A9: w in ker (Mx2Tran (A,b1,b2)) by STRUCT_0:def_5; then w in V1 by VECTSP_4:9; then reconsider W = w as Vector of V1 by STRUCT_0:def_5; ((Mx2Tran (AI,b1,BB)) | (ker (Mx2Tran (A,b1,b2)))) . w = (Mx2Tran (AI,b1,BB)) . w by A5, FUNCT_1:47 .= ((id V1) . W) |-- b1 by Th47 .= W |-- b1 by FUNCT_1:17 ; then ((Mx2Tran (AI,b1,BB)) | (ker (Mx2Tran (A,b1,b2)))) . w in Space_of_Solutions_of (A @) by A1, A2, A9, Th41; hence x in the carrier of (Space_of_Solutions_of (A @)) by A8, STRUCT_0:def_5; ::_thesis: verum end; ( len A = len b1 & width A = len b2 ) by A1, MATRIX_1:23; then A10: width (A @) = len b1 by A2, MATRIX_2:10; A11: Mx2Tran (AI,b1,BB) is one-to-one by A4, Th44; dim ((len b1) -VectSp_over K) = len b1 by MATRIX13:112 .= dim V1 by Th21 .= rank (Mx2Tran (AI,b1,BB)) by A11, RANKNULL:45 .= dim (im (Mx2Tran (AI,b1,BB))) ; then A12: (Omega). ((len b1) -VectSp_over K) = (Omega). (im (Mx2Tran (AI,b1,BB))) by VECTSP_9:28; the carrier of (Space_of_Solutions_of (A @)) c= the carrier of (im ((Mx2Tran (AI,b1,BB)) | (ker (Mx2Tran (A,b1,b2))))) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in the carrier of (Space_of_Solutions_of (A @)) or x in the carrier of (im ((Mx2Tran (AI,b1,BB)) | (ker (Mx2Tran (A,b1,b2))))) ) assume A13: x in the carrier of (Space_of_Solutions_of (A @)) ; ::_thesis: x in the carrier of (im ((Mx2Tran (AI,b1,BB)) | (ker (Mx2Tran (A,b1,b2))))) the carrier of (Space_of_Solutions_of (A @)) c= the carrier of ((len b1) -VectSp_over K) by A10, VECTSP_4:def_2; then reconsider v = x as Element of ((len b1) -VectSp_over K) by A13; A14: v in Space_of_Solutions_of (A @) by A13, STRUCT_0:def_5; v in im (Mx2Tran (AI,b1,BB)) by A12, STRUCT_0:def_5; then consider w being Element of V1 such that A15: v = (Mx2Tran (AI,b1,BB)) . w by RANKNULL:13; (Mx2Tran (AI,b1,BB)) . w = ((id V1) . w) |-- b1 by Th47 .= w |-- b1 by FUNCT_1:17 ; then w in ker (Mx2Tran (A,b1,b2)) by A1, A2, A15, A14, Th41; then reconsider W = w as Element of (ker (Mx2Tran (A,b1,b2))) by STRUCT_0:def_5; v = ((Mx2Tran (AI,b1,BB)) | (ker (Mx2Tran (A,b1,b2)))) . W by A5, A15, FUNCT_1:47; then v in im ((Mx2Tran (AI,b1,BB)) | (ker (Mx2Tran (A,b1,b2)))) by RANKNULL:13; hence x in the carrier of (im ((Mx2Tran (AI,b1,BB)) | (ker (Mx2Tran (A,b1,b2))))) by STRUCT_0:def_5; ::_thesis: verum end; then the carrier of (Space_of_Solutions_of (A @)) = the carrier of (im ((Mx2Tran (AI,b1,BB)) | (ker (Mx2Tran (A,b1,b2))))) by A6, XBOOLE_0:def_10; then A16: im ((Mx2Tran (AI,b1,BB)) | (ker (Mx2Tran (A,b1,b2)))) = Space_of_Solutions_of (A @) by A10, VECTSP_4:29; Mx2Tran (AI,b1,BB) is one-to-one by A4, Th44; then (Mx2Tran (AI,b1,BB)) | (ker (Mx2Tran (A,b1,b2))) is one-to-one by FUNCT_1:52; hence nullity (Mx2Tran (A,b1,b2)) = rank ((Mx2Tran (AI,b1,BB)) | (ker (Mx2Tran (A,b1,b2)))) by RANKNULL:45 .= (len b1) - (the_rank_of (A @)) by A1, A10, A16, MATRIX15:68 .= (len b1) - (the_rank_of A) by MATRIX13:84 ; ::_thesis: verum end; begin theorem Th48: :: MATRLIN2:48 for K being Field for V1, V2 being finite-dimensional VectSp of K for b1 being OrdBasis of V1 for b2 being OrdBasis of V2 for f being linear-transformation of V1,V2 holds rank f = the_rank_of (AutMt (f,b1,b2)) proof let K be Field; ::_thesis: for V1, V2 being finite-dimensional VectSp of K for b1 being OrdBasis of V1 for b2 being OrdBasis of V2 for f being linear-transformation of V1,V2 holds rank f = the_rank_of (AutMt (f,b1,b2)) let V1, V2 be finite-dimensional VectSp of K; ::_thesis: for b1 being OrdBasis of V1 for b2 being OrdBasis of V2 for f being linear-transformation of V1,V2 holds rank f = the_rank_of (AutMt (f,b1,b2)) let b1 be OrdBasis of V1; ::_thesis: for b2 being OrdBasis of V2 for f being linear-transformation of V1,V2 holds rank f = the_rank_of (AutMt (f,b1,b2)) let b2 be OrdBasis of V2; ::_thesis: for f being linear-transformation of V1,V2 holds rank f = the_rank_of (AutMt (f,b1,b2)) let f be linear-transformation of V1,V2; ::_thesis: rank f = the_rank_of (AutMt (f,b1,b2)) set A = AutMt (f,b1,b2); percases ( len b1 = 0 or ( len b1 > 0 & len b2 = 0 ) or ( len b1 > 0 & len b2 > 0 ) ) ; supposeA1: len b1 = 0 ; ::_thesis: rank f = the_rank_of (AutMt (f,b1,b2)) then len (AutMt (f,b1,b2)) = 0 by MATRIX_1:def_2; then ( dim V1 = (rank f) + (nullity f) & the_rank_of (AutMt (f,b1,b2)) = 0 ) by MATRIX13:74, RANKNULL:44; hence rank f = the_rank_of (AutMt (f,b1,b2)) by A1, Th21; ::_thesis: verum end; supposeA2: ( len b1 > 0 & len b2 = 0 ) ; ::_thesis: rank f = the_rank_of (AutMt (f,b1,b2)) then width (AutMt (f,b1,b2)) = 0 by MATRIX_1:23; then A3: the_rank_of (AutMt (f,b1,b2)) = 0 by MATRIX13:74; dim V2 = 0 by A2, Th21; hence rank f = the_rank_of (AutMt (f,b1,b2)) by A3, VECTSP_9:25; ::_thesis: verum end; supposeA4: ( len b1 > 0 & len b2 > 0 ) ; ::_thesis: rank f = the_rank_of (AutMt (f,b1,b2)) A5: (rank f) + (nullity f) = dim V1 by RANKNULL:44 .= len b1 by Th21 ; nullity f = nullity (Mx2Tran ((AutMt (f,b1,b2)),b1,b2)) by Th34 .= (len b1) - (the_rank_of (AutMt (f,b1,b2))) by A4, Lm7 ; hence rank f = the_rank_of (AutMt (f,b1,b2)) by A5; ::_thesis: verum end; end; end; theorem :: MATRLIN2:49 for K being Field for V1, V2 being finite-dimensional VectSp of K for b1 being OrdBasis of V1 for b2 being OrdBasis of V2 for M being Matrix of len b1, len b2,K holds rank (Mx2Tran (M,b1,b2)) = the_rank_of M proof let K be Field; ::_thesis: for V1, V2 being finite-dimensional VectSp of K for b1 being OrdBasis of V1 for b2 being OrdBasis of V2 for M being Matrix of len b1, len b2,K holds rank (Mx2Tran (M,b1,b2)) = the_rank_of M let V1, V2 be finite-dimensional VectSp of K; ::_thesis: for b1 being OrdBasis of V1 for b2 being OrdBasis of V2 for M being Matrix of len b1, len b2,K holds rank (Mx2Tran (M,b1,b2)) = the_rank_of M let b1 be OrdBasis of V1; ::_thesis: for b2 being OrdBasis of V2 for M being Matrix of len b1, len b2,K holds rank (Mx2Tran (M,b1,b2)) = the_rank_of M let b2 be OrdBasis of V2; ::_thesis: for M being Matrix of len b1, len b2,K holds rank (Mx2Tran (M,b1,b2)) = the_rank_of M let M be Matrix of len b1, len b2,K; ::_thesis: rank (Mx2Tran (M,b1,b2)) = the_rank_of M thus rank (Mx2Tran (M,b1,b2)) = the_rank_of (AutMt ((Mx2Tran (M,b1,b2)),b1,b2)) by Th48 .= the_rank_of M by Th36 ; ::_thesis: verum end; theorem :: MATRLIN2:50 for K being Field for V1, V2 being finite-dimensional VectSp of K for b1 being OrdBasis of V1 for b2 being OrdBasis of V2 for f being linear-transformation of V1,V2 st dim V1 = dim V2 holds ( not ker f is trivial iff Det (AutEqMt (f,b1,b2)) = 0. K ) proof let K be Field; ::_thesis: for V1, V2 being finite-dimensional VectSp of K for b1 being OrdBasis of V1 for b2 being OrdBasis of V2 for f being linear-transformation of V1,V2 st dim V1 = dim V2 holds ( not ker f is trivial iff Det (AutEqMt (f,b1,b2)) = 0. K ) let V1, V2 be finite-dimensional VectSp of K; ::_thesis: for b1 being OrdBasis of V1 for b2 being OrdBasis of V2 for f being linear-transformation of V1,V2 st dim V1 = dim V2 holds ( not ker f is trivial iff Det (AutEqMt (f,b1,b2)) = 0. K ) let b1 be OrdBasis of V1; ::_thesis: for b2 being OrdBasis of V2 for f being linear-transformation of V1,V2 st dim V1 = dim V2 holds ( not ker f is trivial iff Det (AutEqMt (f,b1,b2)) = 0. K ) let b2 be OrdBasis of V2; ::_thesis: for f being linear-transformation of V1,V2 st dim V1 = dim V2 holds ( not ker f is trivial iff Det (AutEqMt (f,b1,b2)) = 0. K ) let f be linear-transformation of V1,V2; ::_thesis: ( dim V1 = dim V2 implies ( not ker f is trivial iff Det (AutEqMt (f,b1,b2)) = 0. K ) ) assume A1: dim V1 = dim V2 ; ::_thesis: ( not ker f is trivial iff Det (AutEqMt (f,b1,b2)) = 0. K ) set A = AutEqMt (f,b1,b2); dim V2 = len b2 by Th21; then A2: AutEqMt (f,b1,b2) = AutMt (f,b1,b2) by A1, Def2, Th21; A3: dim V1 = (rank f) + (nullity f) by RANKNULL:44; A4: ( len b1 = dim V1 & rank f = the_rank_of (AutMt (f,b1,b2)) ) by Th21, Th48; hereby ::_thesis: ( Det (AutEqMt (f,b1,b2)) = 0. K implies not ker f is trivial ) assume not ker f is trivial ; ::_thesis: Det (AutEqMt (f,b1,b2)) = 0. K then rank f <> dim V1 by A3, Th42; hence Det (AutEqMt (f,b1,b2)) = 0. K by A4, A2, MATRIX13:83; ::_thesis: verum end; assume Det (AutEqMt (f,b1,b2)) = 0. K ; ::_thesis: not ker f is trivial then dim (ker f) <> 0 by A4, A2, A3, MATRIX13:83; hence not ker f is trivial by Th42; ::_thesis: verum end; Lm8: for K being Field for V1, V2, V3 being finite-dimensional VectSp of K for f being linear-transformation of V1,V2 for g being Function of V2,V3 holds g * f = (g | (im f)) * f proof let K be Field; ::_thesis: for V1, V2, V3 being finite-dimensional VectSp of K for f being linear-transformation of V1,V2 for g being Function of V2,V3 holds g * f = (g | (im f)) * f let V1, V2, V3 be finite-dimensional VectSp of K; ::_thesis: for f being linear-transformation of V1,V2 for g being Function of V2,V3 holds g * f = (g | (im f)) * f let f be linear-transformation of V1,V2; ::_thesis: for g being Function of V2,V3 holds g * f = (g | (im f)) * f let g be Function of V2,V3; ::_thesis: g * f = (g | (im f)) * f dom f = [#] V1 by FUNCT_2:def_1; then [#] (im f) = f .: (dom f) by RANKNULL:def_2 .= rng f by RELAT_1:113 ; hence (g | (im f)) * f = (g * (id (rng f))) * f by RELAT_1:65 .= g * ((id (rng f)) * f) by RELAT_1:36 .= g * f by RELAT_1:54 ; ::_thesis: verum end; theorem :: MATRLIN2:51 for K being Field for V1, V2, V3 being finite-dimensional VectSp of K for f being linear-transformation of V1,V2 for g being linear-transformation of V2,V3 st g | (im f) is one-to-one holds ( rank (g * f) = rank f & nullity (g * f) = nullity f ) proof let K be Field; ::_thesis: for V1, V2, V3 being finite-dimensional VectSp of K for f being linear-transformation of V1,V2 for g being linear-transformation of V2,V3 st g | (im f) is one-to-one holds ( rank (g * f) = rank f & nullity (g * f) = nullity f ) let V1, V2, V3 be finite-dimensional VectSp of K; ::_thesis: for f being linear-transformation of V1,V2 for g being linear-transformation of V2,V3 st g | (im f) is one-to-one holds ( rank (g * f) = rank f & nullity (g * f) = nullity f ) let f be linear-transformation of V1,V2; ::_thesis: for g being linear-transformation of V2,V3 st g | (im f) is one-to-one holds ( rank (g * f) = rank f & nullity (g * f) = nullity f ) let g be linear-transformation of V2,V3; ::_thesis: ( g | (im f) is one-to-one implies ( rank (g * f) = rank f & nullity (g * f) = nullity f ) ) assume A1: g | (im f) is one-to-one ; ::_thesis: ( rank (g * f) = rank f & nullity (g * f) = nullity f ) the carrier of (im (g * f)) = [#] (im (g * f)) .= (g * f) .: ([#] V1) by RANKNULL:def_2 .= ((g | (im f)) * f) .: ([#] V1) by Lm8 .= (g | (im f)) .: (f .: ([#] V1)) by RELAT_1:126 .= (g | (im f)) .: ([#] (im f)) by RANKNULL:def_2 .= [#] (im (g | (im f))) by RANKNULL:def_2 .= the carrier of (im (g | (im f))) ; then A2: rank (g * f) = rank (g | (im f)) by VECTSP_4:29 .= rank f by A1, RANKNULL:45 ; (nullity f) + (rank f) = dim V1 by RANKNULL:44 .= (nullity (g * f)) + (rank (g * f)) by RANKNULL:44 ; hence ( rank (g * f) = rank f & nullity (g * f) = nullity f ) by A2; ::_thesis: verum end;