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AA2 Chap 19

Vector Space A vector space V over a field F, is an Abelian group under addition and closed over scalar multiplication. The elements of F are called SCALARS The elements of V are called VECTORS
Examples of Vector Spaces R^n is a vector space over R M(2x2) is a vector space over R C is a vector space over R
Basis for V A BASIS for V is a set of vectors that are linearly independent and span V.
linearly independent v1,v2,...,vn are linearly independent if the only solution to c1v1+c2v2+...+cnvn=0, is ci=0
span v1,v2,...,vn SPAN V if for all v in V there exists ci such that c1v1+c2v2+...+cnvn=v
A basis is a... minimal spanning set
A basis is a..... maximal linearly independent set
Theorem about Vector spaces and Bases Every vector space has a basis Proof: Zorn's Lemma
Vector Space letters For all v in V, a in F, av in V, 1) a(v+w)=av+aw 2)(a+b)v=av+bv 3)(ab)v=a(bv) 4)1*v=v
Subspace U is a SUBSPACE of V if U is a subset that is also a vector space over F.
Subspace spanned Let v1,v2,...,vn be in V, then <v1,v2,...,vn>={a1v1+a2v2+...+anvn|ai is in F} is called a subspace spanned by v1,v2,...vn.
Linear combination a1v1+a2v2+...+anvn is called a linear combination of v1,v2,...,vn if V=<v1,v2,...,vn>, then {v1,v2,...vn} spans V. {v1,v2,...,vn} is a spanning set for V.
Linearly independent v1,v2,...,vn are LINEARLY INDEPENDENT over F if there exists a1,a2,...,an in F (not all 0) such that a1v1+a2v2+...+anvn=0
Created by: csebald