:: 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;