:: by JingChao Chen

::

:: Received July 1, 1997

:: Copyright (c) 1997-2012 Association of Mizar Users

begin

theorem Th1: :: RLVECT_5:1

for V being RealLinearSpace

for KL1, KL2 being Linear_Combination of V

for X being Subset of V st X is linearly-independent & Carrier KL1 c= X & Carrier KL2 c= X & Sum KL1 = Sum KL2 holds

KL1 = KL2

for KL1, KL2 being Linear_Combination of V

for X being Subset of V st X is linearly-independent & Carrier KL1 c= X & Carrier KL2 c= X & Sum KL1 = Sum KL2 holds

KL1 = KL2

proof end;

theorem Th2: :: RLVECT_5:2

for V being RealLinearSpace

for A being Subset of V st A is linearly-independent holds

ex I being Basis of V st A c= I

for A being Subset of V st A is linearly-independent holds

ex I being Basis of V st A c= I

proof end;

theorem Th3: :: RLVECT_5:3

for V being RealLinearSpace

for L being Linear_Combination of V

for x being VECTOR of V holds

( x in Carrier L iff ex v being VECTOR of V st

( x = v & L . v <> 0 ) )

for L being Linear_Combination of V

for x being VECTOR of V holds

( x in Carrier L iff ex v being VECTOR of V st

( x = v & L . v <> 0 ) )

proof end;

Lm1: for X, x being set st x in X holds

(X \ {x}) \/ {x} = X

proof end;

:: More On Linear Combinations

theorem Th4: :: RLVECT_5:4

for V being RealLinearSpace

for L being Linear_Combination of V

for F, G being FinSequence of the carrier of V

for P being Permutation of (dom F) st G = F * P holds

Sum (L (#) F) = Sum (L (#) G)

for L being Linear_Combination of V

for F, G being FinSequence of the carrier of V

for P being Permutation of (dom F) st G = F * P holds

Sum (L (#) F) = Sum (L (#) G)

proof end;

theorem Th5: :: RLVECT_5:5

for V being RealLinearSpace

for L being Linear_Combination of V

for F being FinSequence of the carrier of V st Carrier L misses rng F holds

Sum (L (#) F) = 0. V

for L being Linear_Combination of V

for F being FinSequence of the carrier of V st Carrier L misses rng F holds

Sum (L (#) F) = 0. V

proof end;

theorem Th6: :: RLVECT_5:6

for V being RealLinearSpace

for F being FinSequence of the carrier of V st F is one-to-one holds

for L being Linear_Combination of V st Carrier L c= rng F holds

Sum (L (#) F) = Sum L

for F being FinSequence of the carrier of V st F is one-to-one holds

for L being Linear_Combination of V st Carrier L c= rng F holds

Sum (L (#) F) = Sum L

proof end;

theorem Th7: :: RLVECT_5:7

for V being RealLinearSpace

for L being Linear_Combination of V

for F being FinSequence of the carrier of V ex K being Linear_Combination of V st

( Carrier K = (rng F) /\ (Carrier L) & L (#) F = K (#) F )

for L being Linear_Combination of V

for F being FinSequence of the carrier of V ex K being Linear_Combination of V st

( Carrier K = (rng F) /\ (Carrier L) & L (#) F = K (#) F )

proof end;

theorem Th8: :: RLVECT_5:8

for V being RealLinearSpace

for L being Linear_Combination of V

for A being Subset of V

for F being FinSequence of the carrier of V st rng F c= the carrier of (Lin A) holds

ex K being Linear_Combination of A st Sum (L (#) F) = Sum K

for L being Linear_Combination of V

for A being Subset of V

for F being FinSequence of the carrier of V st rng F c= the carrier of (Lin A) holds

ex K being Linear_Combination of A st Sum (L (#) F) = Sum K

proof end;

theorem Th9: :: RLVECT_5:9

for V being RealLinearSpace

for L being Linear_Combination of V

for A being Subset of V st Carrier L c= the carrier of (Lin A) holds

ex K being Linear_Combination of A st Sum L = Sum K

for L being Linear_Combination of V

for A being Subset of V st Carrier L c= the carrier of (Lin A) holds

ex K being Linear_Combination of A st Sum L = Sum K

proof end;

theorem Th10: :: RLVECT_5:10

for V being RealLinearSpace

for W being Subspace of V

for L being Linear_Combination of V st Carrier L c= the carrier of W holds

for K being Linear_Combination of W st K = L | the carrier of W holds

( Carrier L = Carrier K & Sum L = Sum K )

for W being Subspace of V

for L being Linear_Combination of V st Carrier L c= the carrier of W holds

for K being Linear_Combination of W st K = L | the carrier of W holds

( Carrier L = Carrier K & Sum L = Sum K )

proof end;

theorem Th11: :: RLVECT_5:11

for V being RealLinearSpace

for W being Subspace of V

for K being Linear_Combination of W ex L being Linear_Combination of V st

( Carrier K = Carrier L & Sum K = Sum L )

for W being Subspace of V

for K being Linear_Combination of W ex L being Linear_Combination of V st

( Carrier K = Carrier L & Sum K = Sum L )

proof end;

theorem Th12: :: RLVECT_5:12

for V being RealLinearSpace

for W being Subspace of V

for L being Linear_Combination of V st Carrier L c= the carrier of W holds

ex K being Linear_Combination of W st

( Carrier K = Carrier L & Sum K = Sum L )

for W being Subspace of V

for L being Linear_Combination of V st Carrier L c= the carrier of W holds

ex K being Linear_Combination of W st

( Carrier K = Carrier L & Sum K = Sum L )

proof end;

:: More On Linear Independence & Basis

theorem Th14: :: RLVECT_5:14

for V being RealLinearSpace

for W being Subspace of V

for A being Subset of W st A is linearly-independent holds

A is linearly-independent Subset of V

for W being Subspace of V

for A being Subset of W st A is linearly-independent holds

A is linearly-independent Subset of V

proof end;

theorem Th15: :: RLVECT_5:15

for V being RealLinearSpace

for W being Subspace of V

for A being Subset of V st A is linearly-independent & A c= the carrier of W holds

A is linearly-independent Subset of W

for W being Subspace of V

for A being Subset of V st A is linearly-independent & A c= the carrier of W holds

A is linearly-independent Subset of W

proof end;

theorem Th16: :: RLVECT_5:16

for V being RealLinearSpace

for W being Subspace of V

for A being Basis of W ex B being Basis of V st A c= B

for W being Subspace of V

for A being Basis of W ex B being Basis of V st A c= B

proof end;

theorem Th17: :: RLVECT_5:17

for V being RealLinearSpace

for A being Subset of V st A is linearly-independent holds

for v being VECTOR of V st v in A holds

for B being Subset of V st B = A \ {v} holds

not v in Lin B

for A being Subset of V st A is linearly-independent holds

for v being VECTOR of V st v in A holds

for B being Subset of V st B = A \ {v} holds

not v in Lin B

proof end;

theorem Th18: :: RLVECT_5:18

for V being RealLinearSpace

for I being Basis of V

for A being non empty Subset of V st A misses I holds

for B being Subset of V st B = I \/ A holds

B is linearly-dependent

for I being Basis of V

for A being non empty Subset of V st A misses I holds

for B being Subset of V st B = I \/ A holds

B is linearly-dependent

proof end;

theorem Th19: :: RLVECT_5:19

for V being RealLinearSpace

for W being Subspace of V

for A being Subset of V st A c= the carrier of W holds

Lin A is Subspace of W

for W being Subspace of V

for A being Subset of V st A c= the carrier of W holds

Lin A is Subspace of W

proof end;

theorem Th20: :: RLVECT_5:20

for V being RealLinearSpace

for W being Subspace of V

for A being Subset of V

for B being Subset of W st A = B holds

Lin A = Lin B

for W being Subspace of V

for A being Subset of V

for B being Subset of W st A = B holds

Lin A = Lin B

proof end;

begin

:: Exchange Lemma

theorem Th21: :: RLVECT_5:21

for V being RealLinearSpace

for A, B being finite Subset of V

for v being VECTOR of V st v in Lin (A \/ B) & not v in Lin B holds

ex w being VECTOR of V st

( w in A & w in Lin (((A \/ B) \ {w}) \/ {v}) )

for A, B being finite Subset of V

for v being VECTOR of V st v in Lin (A \/ B) & not v in Lin B holds

ex w being VECTOR of V st

( w in A & w in Lin (((A \/ B) \ {w}) \/ {v}) )

proof end;

:: Steinitz Theorem

theorem Th22: :: RLVECT_5:22

for V being RealLinearSpace

for A, B being finite Subset of V st RLSStruct(# the carrier of V, the ZeroF of V, the U5 of V, the Mult of V #) = Lin A & B is linearly-independent holds

( card B <= card A & ex C being finite Subset of V st

( C c= A & card C = (card A) - (card B) & RLSStruct(# the carrier of V, the ZeroF of V, the U5 of V, the Mult of V #) = Lin (B \/ C) ) )

for A, B being finite Subset of V st RLSStruct(# the carrier of V, the ZeroF of V, the U5 of V, the Mult of V #) = Lin A & B is linearly-independent holds

( card B <= card A & ex C being finite Subset of V st

( C c= A & card C = (card A) - (card B) & RLSStruct(# the carrier of V, the ZeroF of V, the U5 of V, the Mult of V #) = Lin (B \/ C) ) )

proof end;

begin

definition

let V be RealLinearSpace;

end;
attr V is finite-dimensional means :Def1: :: RLVECT_5:def 1

ex A being finite Subset of V st A is Basis of V;

ex A being finite Subset of V st A is Basis of V;

:: deftheorem Def1 defines finite-dimensional RLVECT_5:def 1 :

for V being RealLinearSpace holds

( V is finite-dimensional iff ex A being finite Subset of V st A is Basis of V );

for V being RealLinearSpace holds

( V is finite-dimensional iff ex A being finite Subset of V st A is Basis of V );

registration

ex b_{1} being RealLinearSpace st

( b_{1} is strict & b_{1} is finite-dimensional )
end;

cluster non empty V83() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional for RLSStruct ;

existence ex b

( b

proof end;

theorem Th23: :: RLVECT_5:23

for V being RealLinearSpace st V is finite-dimensional holds

for I being Basis of V holds I is finite

for I being Basis of V holds I is finite

proof end;

theorem :: RLVECT_5:24

for V being RealLinearSpace st V is finite-dimensional holds

for A being Subset of V st A is linearly-independent holds

A is finite

for A being Subset of V st A is linearly-independent holds

A is finite

proof end;

theorem Th25: :: RLVECT_5:25

for V being RealLinearSpace st V is finite-dimensional holds

for A, B being Basis of V holds card A = card B

for A, B being Basis of V holds card A = card B

proof end;

theorem Th27: :: RLVECT_5:27

for V being RealLinearSpace

for W being Subspace of V st V is finite-dimensional holds

W is finite-dimensional

for W being Subspace of V st V is finite-dimensional holds

W is finite-dimensional

proof end;

registration

let V be RealLinearSpace;

ex b_{1} being Subspace of V st

( b_{1} is finite-dimensional & b_{1} is strict )

end;
cluster non empty V83() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional for Subspace of V;

existence ex b

( b

proof end;

registration

let V be finite-dimensional RealLinearSpace;

correctness

coherence

for b_{1} being Subspace of V holds b_{1} is finite-dimensional ;

by Th27;

end;
correctness

coherence

for b

by Th27;

registration

let V be finite-dimensional RealLinearSpace;

ex b_{1} being Subspace of V st b_{1} is strict

end;
cluster non empty V83() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional for Subspace of V;

existence ex b

proof end;

begin

definition

let V be RealLinearSpace;

assume A1: V is finite-dimensional ;

ex b_{1} being Element of NAT st

for I being Basis of V holds b_{1} = card I

for b_{1}, b_{2} being Element of NAT st ( for I being Basis of V holds b_{1} = card I ) & ( for I being Basis of V holds b_{2} = card I ) holds

b_{1} = b_{2}

end;
assume A1: V is finite-dimensional ;

func dim V -> Element of NAT means :Def2: :: RLVECT_5:def 2

for I being Basis of V holds it = card I;

existence for I being Basis of V holds it = card I;

ex b

for I being Basis of V holds b

proof end;

uniqueness for b

b

proof end;

:: deftheorem Def2 defines dim RLVECT_5:def 2 :

for V being RealLinearSpace st V is finite-dimensional holds

for b_{2} being Element of NAT holds

( b_{2} = dim V iff for I being Basis of V holds b_{2} = card I );

for V being RealLinearSpace st V is finite-dimensional holds

for b

( b

theorem Th29: :: RLVECT_5:29

for V being finite-dimensional RealLinearSpace

for A being Subset of V st A is linearly-independent holds

card A = dim (Lin A)

for A being Subset of V st A is linearly-independent holds

card A = dim (Lin A)

proof end;

theorem :: RLVECT_5:31

for V being finite-dimensional RealLinearSpace

for W being Subspace of V holds

( dim V = dim W iff (Omega). V = (Omega). W )

for W being Subspace of V holds

( dim V = dim W iff (Omega). V = (Omega). W )

proof end;

theorem :: RLVECT_5:33

for V being finite-dimensional RealLinearSpace holds

( dim V = 1 iff ex v being VECTOR of V st

( v <> 0. V & (Omega). V = Lin {v} ) )

( dim V = 1 iff ex v being VECTOR of V st

( v <> 0. V & (Omega). V = Lin {v} ) )

proof end;

theorem :: RLVECT_5:34

for V being finite-dimensional RealLinearSpace holds

( dim V = 2 iff ex u, v being VECTOR of V st

( u <> v & {u,v} is linearly-independent & (Omega). V = Lin {u,v} ) )

( dim V = 2 iff ex u, v being VECTOR of V st

( u <> v & {u,v} is linearly-independent & (Omega). V = Lin {u,v} ) )

proof end;

theorem Th35: :: RLVECT_5:35

for V being finite-dimensional RealLinearSpace

for W1, W2 being Subspace of V holds (dim (W1 + W2)) + (dim (W1 /\ W2)) = (dim W1) + (dim W2)

for W1, W2 being Subspace of V holds (dim (W1 + W2)) + (dim (W1 /\ W2)) = (dim W1) + (dim W2)

proof end;

theorem :: RLVECT_5:36

for V being finite-dimensional RealLinearSpace

for W1, W2 being Subspace of V holds dim (W1 /\ W2) >= ((dim W1) + (dim W2)) - (dim V)

for W1, W2 being Subspace of V holds dim (W1 /\ W2) >= ((dim W1) + (dim W2)) - (dim V)

proof end;

theorem :: RLVECT_5:37

for V being finite-dimensional RealLinearSpace

for W1, W2 being Subspace of V st V is_the_direct_sum_of W1,W2 holds

dim V = (dim W1) + (dim W2)

for W1, W2 being Subspace of V st V is_the_direct_sum_of W1,W2 holds

dim V = (dim W1) + (dim W2)

proof end;

Lm2: for n being Element of NAT

for V being finite-dimensional RealLinearSpace st n <= dim V holds

ex W being strict Subspace of V st dim W = n

proof end;

theorem :: RLVECT_5:38

definition

let V be finite-dimensional RealLinearSpace;

let n be Element of NAT ;

ex b_{1} being set st

for x being set holds

( x in b_{1} iff ex W being strict Subspace of V st

( W = x & dim W = n ) )

for b_{1}, b_{2} being set st ( for x being set holds

( x in b_{1} iff ex W being strict Subspace of V st

( W = x & dim W = n ) ) ) & ( for x being set holds

( x in b_{2} iff ex W being strict Subspace of V st

( W = x & dim W = n ) ) ) holds

b_{1} = b_{2}

end;
let n be Element of NAT ;

func n Subspaces_of V -> set means :Def3: :: RLVECT_5:def 3

for x being set holds

( x in it iff ex W being strict Subspace of V st

( W = x & dim W = n ) );

existence for x being set holds

( x in it iff ex W being strict Subspace of V st

( W = x & dim W = n ) );

ex b

for x being set holds

( x in b

( W = x & dim W = n ) )

proof end;

uniqueness for b

( x in b

( W = x & dim W = n ) ) ) & ( for x being set holds

( x in b

( W = x & dim W = n ) ) ) holds

b

proof end;

:: deftheorem Def3 defines Subspaces_of RLVECT_5:def 3 :

for V being finite-dimensional RealLinearSpace

for n being Element of NAT

for b_{3} being set holds

( b_{3} = n Subspaces_of V iff for x being set holds

( x in b_{3} iff ex W being strict Subspace of V st

( W = x & dim W = n ) ) );

for V being finite-dimensional RealLinearSpace

for n being Element of NAT

for b

( b

( x in b

( W = x & dim W = n ) ) );

theorem :: RLVECT_5:39

for n being Element of NAT

for V being finite-dimensional RealLinearSpace st n <= dim V holds

not n Subspaces_of V is empty

for V being finite-dimensional RealLinearSpace st n <= dim V holds

not n Subspaces_of V is empty

proof end;

theorem :: RLVECT_5:40

for n being Element of NAT

for V being finite-dimensional RealLinearSpace st dim V < n holds

n Subspaces_of V = {}

for V being finite-dimensional RealLinearSpace st dim V < n holds

n Subspaces_of V = {}

proof end;

theorem :: RLVECT_5:41

for n being Element of NAT

for V being finite-dimensional RealLinearSpace

for W being Subspace of V holds n Subspaces_of W c= n Subspaces_of V

for V being finite-dimensional RealLinearSpace

for W being Subspace of V holds n Subspaces_of W c= n Subspaces_of V

proof end;