Volume 2, 1990

University of Bialystok

Copyright (c) 1990 Association of Mizar Users

### The abstract of the Mizar article:

### Basis of Vector Space

**by****Wojciech A. Trybulec**- Received July 27, 1990
- 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;

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