Journal of Formalized Mathematics
Volume 2, 1990
University of Bialystok
Copyright (c) 1990 Association of Mizar Users

### Basis of Vector Space

by
Wojciech A. Trybulec

MML identifier: VECTSP_7
[ Mizar article, MML identifier index ]

```environ

vocabulary VECTSP_1, FINSET_1, RLVECT_2, FUNCT_1, RLVECT_3, RLVECT_1, BOOLE,
FUNCT_2, ARYTM_1, RELAT_1, RLSUB_1, ZFMISC_1, TARSKI, ORDERS_1, HAHNBAN;
notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, RELAT_1, FUNCT_1, FUNCT_2,
FRAENKEL, FINSET_1, STRUCT_0, ORDINAL1, ORDERS_1, RLVECT_1, VECTSP_1,
RLVECT_2, VECTSP_4, VECTSP_5, VECTSP_6;
constructors ORDERS_1, RLVECT_2, VECTSP_5, VECTSP_6, MEMBERED, XBOOLE_0;
clusters VECTSP_1, VECTSP_4, RLVECT_2, STRUCT_0, RELSET_1, SUBSET_1, MEMBERED,
ZFMISC_1, XBOOLE_0;
requirements SUBSET, BOOLE;

begin

reserve x,y,X,Y,Z for set;
reserve GF for Field;
reserve a,b for Element of GF;
reserve V for VectSp of GF;
reserve v,v1,v2,u for Vector of V;
reserve A,B,C for Subset of V;
reserve T for finite Subset of V;
reserve l for Linear_Combination of A;
reserve f,g for Function of the carrier of V, the carrier of GF;

definition let GF; let V;
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) = {};
antonym IT is linearly-dependent;
end;

canceled;

theorem :: VECTSP_7:2
A c= B & B is linearly-independent implies A is linearly-independent;

theorem :: VECTSP_7:3
A is linearly-independent implies not 0.V in A;

theorem :: VECTSP_7:4
{}(the carrier of V) is linearly-independent;

theorem :: VECTSP_7:5
{v} is linearly-independent iff v <> 0.V;

theorem :: VECTSP_7:6
{v1,v2} is linearly-independent implies v1 <> 0.V & v2 <> 0.V;

theorem :: VECTSP_7:7
{v,0.V} is linearly-dependent & {0.V,v} is linearly-dependent;

theorem :: VECTSP_7:8
v1 <> v2 & {v1,v2} is linearly-independent iff
v2 <> 0.V & for a holds v1 <> a * v2;

theorem :: VECTSP_7:9
v1 <> v2 & {v1,v2} is linearly-independent iff
for a,b st a * v1 + b * v2 = 0.V holds a = 0.GF & b = 0.GF;

definition let GF; let V; let A;
func Lin(A) -> strict Subspace of V means
:: VECTSP_7:def 2

the carrier of it = {Sum(l) : not contradiction};
end;

canceled 2;

theorem :: VECTSP_7:12
x in Lin(A) iff ex l st x = Sum(l);

theorem :: VECTSP_7:13
x in A implies x in Lin(A);

reserve l0 for Linear_Combination of {}(the carrier of V);

theorem :: VECTSP_7:14
Lin({}(the carrier of V)) = (0).V;

theorem :: VECTSP_7:15
Lin(A) = (0).V implies A = {} or A = {0.V};

theorem :: VECTSP_7:16
for W being strict Subspace of V st A = the carrier of W
holds Lin(A) = W;

theorem :: VECTSP_7:17
for V being strict VectSp of GF, A being Subset of V st
A = the carrier of V holds Lin(A) = V;

theorem :: VECTSP_7:18
A c= B implies Lin(A) is Subspace of Lin(B);

theorem :: VECTSP_7:19
for V being strict VectSp of GF, A,B being Subset of V
st Lin(A) = V & A c= B holds Lin(B) = V;

theorem :: VECTSP_7:20
Lin(A \/ B) = Lin(A) + Lin(B);

theorem :: VECTSP_7:21
Lin(A /\ B) is Subspace of Lin(A) /\ Lin(B);

theorem :: VECTSP_7:22
for V being VectSp of GF, 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) = the VectSpStr of V;

theorem :: VECTSP_7:23
Lin(A) = V implies
ex B st B c= A & B is linearly-independent & Lin(B) = V;

definition let GF; let V be VectSp of GF;
mode Basis of V -> Subset of V means
:: VECTSP_7:def 3
it is linearly-independent & Lin(it) = the VectSpStr of V;
end;

canceled 3;

theorem :: VECTSP_7:27
for V being VectSp of GF, A being Subset of V st
A is linearly-independent
holds ex I being Basis of V st A c= I;

theorem :: VECTSP_7:28
for V being VectSp of GF, A being Subset of V st Lin(A) = V
holds ex I being Basis of V st I c= A;
```