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Problems and solutions - MIT Mathematics

CHAPTER 5 Problems and solutions1. Problems Chapter from first principles that ifVis a vector space (overRorC) then for any setXthe space( )F(X;V) ={u:X V}is a linear space over the same field, with pointwise operations . a vector space andS Vis a subset which is closedunder addition and scalar multiplication:( )v1,v2 S, K= v1+v2 Sand v1 SthenSis a vector space as well (called of course a subspace). Vbe a linear subspace of a vector space show that therelation onV( )v1 v2 v1 v2 Sis an equivalence relation and that the set of equivalence classes, denoted usuallyV/S,is a vector space in a natural case you do not know it, go through the basic theory offinite-dimensional vector spaces.

To say that this is an equivalence relation means that symmetry and transitivity hold. Since Sis a subspace, v2Simplies v2Sso v 1 ˘v 2 =)v 1 v 2 2S=)v 2 v 1 2S=)v 2 ˘v 1: Similarly, since it is also possible to add and remain in S v 1 ˘v 2; v 2 ˘v 3 =)v 1 v 2; v 2 v 3 2S=)v 1 v 3 2S=)v 1 ˘v 3: So this is an equivalence relation and the ...

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