:: VECTSP_7 semantic presentation

definition

let GF be Field;
let V be VectSp of GF;
let IT be Subset of V;
attr IT is linearly-independent means :Def1: :: VECTSP_7:def 1
for l being Linear_Combination of IT st Sum l = 0. V holds
Carrier l = {} ;

end;

:: deftheorem Def1 defines linearly-independent VECTSP_7:def 1 :

notation

let GF be Field;
let V be VectSp of GF;
let IT be Subset of V;
antonym linearly-dependent IT for linearly-independent IT;

end;

theorem :: VECTSP_7:1

canceled;

theorem :: VECTSP_7:2

for GF being Field
for V being VectSp of GF
for A, B being Subset of V st A c= B & B is linearly-independent holds
A is linearly-independent

proof end;

theorem Th3: :: VECTSP_7:3

for GF being Field
for V being VectSp of GF
for A being Subset of V st A is linearly-independent holds
not 0. V in A

proof end;

theorem Th4: :: VECTSP_7:4

for GF being Field
for V being VectSp of GF holds {} the carrier of V is linearly-independent

proof end;

registration

let GF be Field;
let V be VectSp of GF;
cluster linearly-independent Element of bool the carrier of V;
existence

proof end;

end;

theorem :: VECTSP_7:5

for GF being Field
for V being VectSp of GF
for v being Vector of V holds
( {v} is linearly-independent iff v <> 0. V )

proof end;

theorem Th6: :: VECTSP_7:6

for GF being Field
for V being VectSp of GF
for v1, v2 being Vector of V st {v1,v2} is linearly-independent holds
( v1 <> 0. V & v2 <> 0. V )

proof end;

theorem :: VECTSP_7:7

for GF being Field
for V being VectSp of GF
for v being Vector of V holds
( not {v,(0. V)} is linearly-independent & not {(0. V),v} is linearly-independent ) by Th6;

theorem Th8: :: VECTSP_7:8

for GF being Field
for V being VectSp of 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_7:9

for GF being Field
for V being VectSp of 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;

definition

let GF be Field;
let V be VectSp 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

proof end;

uniqueness by VECTSP_4:37;

end;

:: deftheorem Def2 defines Lin VECTSP_7:def 2 :

theorem :: VECTSP_7:10

canceled;

theorem :: VECTSP_7:11

canceled;

theorem Th12: :: VECTSP_7:12

for x being set
for GF being Field
for V being VectSp 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 Th13: :: VECTSP_7:13

for x being set
for GF being Field
for V being VectSp of GF
for A being Subset of V st x in A holds
x in Lin A

proof end;

theorem :: VECTSP_7:14

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

proof end;

theorem :: VECTSP_7:15

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

proof end;

theorem Th16: :: VECTSP_7:16

for GF being Field
for V being VectSp 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_7:17

for GF being Field
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 Th18: :: VECTSP_7:18

for GF being Field
for V being VectSp 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_7:19

for GF being Field
for V being strict VectSp of GF
for A, B being Subset of V st Lin A = V & A c= B holds
Lin B = V

proof end;

theorem :: VECTSP_7:20

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

proof end;

theorem :: VECTSP_7:21

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

proof end;

theorem Th22: :: VECTSP_7:22

for GF being Field
for V being VectSp of GF
for A being Subset of V st A is linearly-independent holds
ex B being Subset of V st
( A c= B & B is linearly-independent & Lin B = VectSpStr(# the carrier of V,the add of V,the Zero of V,the lmult of V #) )

proof end;

theorem Th23: :: VECTSP_7:23

for GF being Field
for V being VectSp of 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;

definition

let GF be Field;
let V be VectSp of GF;
mode Basis of V -> Subset of V means :Def3: :: VECTSP_7:def 3
( it is linearly-independent & Lin it = VectSpStr(# the carrier of V,the add of V,the Zero of V,the lmult of V #) );
existence

proof end;

end;

:: deftheorem Def3 defines Basis VECTSP_7:def 3 :

theorem :: VECTSP_7:24

canceled;

theorem :: VECTSP_7:25

canceled;

theorem :: VECTSP_7:26

canceled;

theorem :: VECTSP_7:27

for GF being Field
for V being VectSp 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_7:28

for GF being Field
for V being VectSp 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;