:: Basis of Vector Space
:: by Wojciech A. Trybulec
::
:: Received July 27, 1990
:: Copyright (c) 1990-2021 Association of Mizar Users

definition

let GF be non empty doubleLoopStr ;
let V be non empty ModuleStr over GF;
let IT be Subset of V;

attr IT is linearly-independent means :: VECTSP_7:def 1
for l being Linear_Combination of IT st Sum l = 0. V holds
Carrier l = {} ;

end;

:: deftheorem defines linearly-independent VECTSP_7:def 1 :
for GF being non empty doubleLoopStr
for V being non empty ModuleStr over GF
for IT being Subset of V holds
( IT is linearly-independent iff for l being Linear_Combination of IT st Sum l = 0. V holds
Carrier l = {} );

notation

let GF be non empty doubleLoopStr ;
let V be non empty ModuleStr over GF;
let IT be Subset of V;
antonym linearly-dependent IT for linearly-independent ;

end;

theorem :: VECTSP_7:1

for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative commutative doubleLoopStr
for V being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF
for A, B being Subset of V st A c= B & B is linearly-independent holds
A is linearly-independent

proof end;

theorem Th2: :: VECTSP_7:2

for R being non degenerated Ring
for V being LeftMod of R
for A being Subset of V st A is linearly-independent holds
not 0. V in A

proof end;

registration

let GF be non empty non degenerated right_complementable almost_left_invertible well-unital distributive Abelian add-associative right_zeroed associative commutative doubleLoopStr ;
let V be non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF;

cluster empty -> linearly-independent for Element of bool the carrier of V;

coherence
for b₁ being Subset of V st b₁ is empty holds
b₁ is linearly-independent

proof end;

end;

registration

let GF be non empty non degenerated right_complementable almost_left_invertible well-unital distributive Abelian add-associative right_zeroed associative commutative doubleLoopStr ;
let V be non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF;

cluster finite linearly-independent for Element of bool the carrier of V;

existence
ex b₁ being Subset of V st
( b₁ is finite & b₁ is linearly-independent )

proof end;

end;

theorem :: VECTSP_7:3

for R being non empty non degenerated right_complementable almost_left_invertible well-unital distributive Abelian add-associative right_zeroed associative commutative doubleLoopStr
for V being LeftMod of R
for v being Vector of V holds
( {v} is linearly-independent iff v <> 0. V )

proof end;

theorem Th4: :: VECTSP_7:4

for GF being non empty non degenerated right_complementable well-unital distributive Abelian add-associative right_zeroed associative commutative doubleLoopStr
for V being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF
for v1, v2 being Vector of V st {v1,v2} is linearly-independent holds
v1 <> 0. V

proof end;

theorem Th5: :: VECTSP_7:5

for GF being non empty non degenerated right_complementable almost_left_invertible well-unital distributive Abelian add-associative right_zeroed associative commutative doubleLoopStr
for V being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF
for v1, v2 being Vector of V holds
( v1 <> v2 & {v1,v2} is linearly-independent iff ( v2 <> 0. V & ( for a being Element of GF holds v1 <> a * v2 ) ) )

proof end;

theorem :: VECTSP_7:6

for GF being non empty non degenerated right_complementable almost_left_invertible well-unital distributive Abelian add-associative right_zeroed associative commutative doubleLoopStr
for V being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF
for v1, v2 being Vector of V holds
( ( v1 <> v2 & {v1,v2} is linearly-independent ) iff for a, b being Element of GF st (a * v1) + (b * v2) = 0. V holds
( a = 0. GF & b = 0. GF ) )

proof end;

TH2: for GF being Ring
for V being LeftMod of GF
for L being Linear_Combination of V
for C being finite Subset of V st Carrier L c= C holds
ex F being FinSequence of V st
( F is one-to-one & rng F = C & Sum L = Sum (L (#) F) )

proof end;

TH3: for GF being Ring
for V being LeftMod of GF
for L being Linear_Combination of V
for a being Scalar of holds Sum (a * L) = a * (Sum L)

proof end;

definition

let GF be Ring;
let V be LeftMod of GF;
let A be Subset of V;

func Lin A -> strict Subspace of V means :Def2: :: VECTSP_7:def 2
the carrier of it = { (Sum l) where l is Linear_Combination of A : verum } ;

existence
ex b₁ being strict Subspace of V st the carrier of b₁ = { (Sum l) where l is Linear_Combination of A : verum }

proof end;

uniqueness
for b₁, b₂ being strict Subspace of V st the carrier of b₁ = { (Sum l) where l is Linear_Combination of A : verum } & the carrier of b₂ = { (Sum l) where l is Linear_Combination of A : verum } holds
b₁ = b₂ by VECTSP_4:29;

end;

:: deftheorem Def2 defines Lin VECTSP_7:def 2 :
for GF being Ring
for V being LeftMod of GF
for A being Subset of V
for b₄ being strict Subspace of V holds
( b₄ = Lin A iff the carrier of b₄ = { (Sum l) where l is Linear_Combination of A : verum } );

theorem Th7: :: VECTSP_7:7

for x being object
for GF being Ring
for V being LeftMod of GF
for A being Subset of V holds
( x in Lin A iff ex l being Linear_Combination of A st x = Sum l )

proof end;

theorem Th8: :: VECTSP_7:8

for x being object
for GF being Ring
for V being LeftMod of GF
for A being Subset of V st x in A holds
x in Lin A

proof end;

theorem Th9: :: VECTSP_7:9

for GF being Ring
for V being LeftMod of GF holds Lin ({} the carrier of V) = (0). V

proof end;

theorem :: VECTSP_7:10

for GF being Ring
for V being LeftMod of GF
for A being Subset of V holds
( not Lin A = (0). V or A = {} or A = {(0. V)} )

proof end;

theorem Th11: :: VECTSP_7:11

for GF being non degenerated Ring
for V being LeftMod of GF
for A being Subset of V
for W being strict Subspace of V st A = the carrier of W holds
Lin A = W

proof end;

theorem :: VECTSP_7:12

for GF being non empty non degenerated right_complementable almost_left_invertible well-unital distributive Abelian add-associative right_zeroed associative commutative doubleLoopStr
for V being strict VectSp of GF
for A being Subset of V st A = the carrier of V holds
Lin A = V

proof end;

theorem Th13: :: VECTSP_7:13

for GF being Ring
for V being LeftMod of GF
for A, B being Subset of V st A c= B holds
Lin A is Subspace of Lin B

proof end;

theorem :: VECTSP_7:14

for GF being non empty non degenerated right_complementable almost_left_invertible well-unital distributive Abelian add-associative right_zeroed associative commutative doubleLoopStr
for V being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF
for A, B being Subset of V st Lin A = V & A c= B holds
Lin B = V

proof end;

theorem :: VECTSP_7:15

for GF being Ring
for V being LeftMod of GF
for A, B being Subset of V holds Lin (A \/ B) = (Lin A) + (Lin B)

proof end;

theorem :: VECTSP_7:16

for GF being Ring
for V being LeftMod of GF
for A, B being Subset of V holds Lin (A /\ B) is Subspace of (Lin A) /\ (Lin B)

proof end;

definition

let GF be non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr ;
let V be LeftMod of GF;
let A be Subset of V;

attr A is base means :Def3: :: VECTSP_7:def 3
( A is linearly-independent & Lin A = ModuleStr(# the carrier of V, the addF of V, the ZeroF of V, the lmult of V #) );

end;

:: deftheorem Def3 defines base VECTSP_7:def 3 :
for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for V being LeftMod of GF
for A being Subset of V holds
( A is base iff ( A is linearly-independent & Lin A = ModuleStr(# the carrier of V, the addF of V, the ZeroF of V, the lmult of V #) ) );

Lm1: for R being Ring
for a being Scalar of R st - a = 0. R holds
a = 0. R

proof end;

Lm2: for R being non degenerated almost_left_invertible Ring
for V being LeftMod of R
for a being Scalar of
for v being Vector of V st a <> 0. R holds
( (a ") * (a * v) = (1. R) * v & ((a ") * a) * v = (1. R) * v )

proof end;

theorem Th17: :: VECTSP_7:17

for R being non degenerated almost_left_invertible Ring
for V being LeftMod of R
for A being Subset of V st A is linearly-independent holds
ex B being Subset of V st
( A c= B & B is base )

proof end;

theorem Th18: :: VECTSP_7:18

for GF being non empty non degenerated right_complementable almost_left_invertible well-unital distributive Abelian add-associative right_zeroed associative commutative doubleLoopStr
for V being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF
for A being Subset of V st Lin A = V holds
ex B being Subset of V st
( B c= A & B is linearly-independent & Lin B = V )

proof end;

Lem0A: for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for V being LeftMod of GF holds {} the carrier of V is linearly-independent

proof end;

registration

let GF be non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr ;
let V be LeftMod of GF;

cluster {} the carrier of V -> linearly-independent ;

coherence
{} the carrier of V is linearly-independent by Lem0A;

end;

registration

let GF be non empty non degenerated right_complementable almost_left_invertible well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr ;
let V be LeftMod of GF;

cluster base for Element of bool the carrier of V;

existence
ex b₁ being Subset of V st b₁ is base

proof end;

end;

definition

let GF be non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr ;
let IT be LeftMod of GF;

attr IT is free means :: VECTSP_7:def 4
ex B being Subset of IT st B is base ;

end;

:: deftheorem defines free VECTSP_7:def 4 :
for GF being non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for IT being LeftMod of GF holds
( IT is free iff ex B being Subset of IT st B is base );

Lem1A: for GF being non empty right_complementable almost_left_invertible well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr
for V being LeftMod of GF holds (0). V is free

proof end;

registration

let GF be non empty right_complementable almost_left_invertible well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr ;
let V be LeftMod of GF;

cluster (0). V -> free ;

coherence
(0). V is free by Lem1A;

end;

registration

let GF be non empty non degenerated right_complementable almost_left_invertible well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr ;

cluster non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed free for ModuleStr over GF;

existence
ex b₁ being LeftMod of GF st
( b₁ is strict & b₁ is free )

proof end;

end;

definition

let GF be non empty non degenerated right_complementable almost_left_invertible well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr ;
let V be LeftMod of GF;
mode Basis of V is base Subset of V;

end;

theorem :: VECTSP_7:19

for GF being non empty non degenerated right_complementable almost_left_invertible well-unital distributive Abelian add-associative right_zeroed associative commutative doubleLoopStr
for V being LeftMod of GF
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 :: VECTSP_7:20

for GF being non empty non degenerated right_complementable almost_left_invertible well-unital distributive Abelian add-associative right_zeroed associative commutative doubleLoopStr
for V being LeftMod of GF
for A being Subset of V st Lin A = V holds
ex I being Basis of V st I c= A

proof end;

:: Additional, copied from MOD_3

theorem :: VECTSP_7:21

for R being Ring
for V being LeftMod of R
for L being Linear_Combination of V
for C being finite Subset of V st Carrier L c= C holds
ex F being FinSequence of V st
( F is one-to-one & rng F = C & Sum L = Sum (L (#) F) ) by TH2;

theorem :: VECTSP_7:22

for R being Ring
for V being LeftMod of R
for L being Linear_Combination of V
for a being Scalar of holds Sum (a * L) = a * (Sum L) by TH3;

theorem :: VECTSP_7:23

for R being non degenerated Ring
for V being LeftMod of R
for v1, v2 being Vector of V st {v1,v2} is linearly-independent holds
( v1 <> 0. V & v2 <> 0. V )

proof end;

theorem :: VECTSP_7:24

for R being domRing
for V being LeftMod of R
for L being Linear_Combination of V
for a being Scalar of R st a <> 0. R holds
Carrier (a * L) = Carrier L

proof end;

theorem Th12: :: VECTSP_7:25

for R being domRing
for V being LeftMod of R
for A being Subset of V
for W being strict Subspace of V st not R is degenerated & A = the carrier of W holds
Lin A = W

proof end;

theorem :: VECTSP_7:26

for R being domRing
for V being strict LeftMod of R
for A being Subset of V st not R is degenerated & A = the carrier of V holds
Lin A = V

proof end;

theorem :: VECTSP_7:27

for R being domRing
for V being strict LeftMod of R
for A, B being Subset of V st Lin A = V & A c= B holds
Lin B = V

proof end;

theorem :: VECTSP_7:28

for R being Ring
for V being LeftMod of R
for v1, v2 being Vector of V st not R is degenerated & {v1,v2} is linearly-independent holds
( v1 <> 0. V & v2 <> 0. V )

proof end;