Transcription of Problems and solutions - MIT Mathematics
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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. Define a vector spaceVto be finite-dimensionalif there is an integerNsuch that anyNelements ofVare linearly dependent ifvi Vfori= 1.
Problems and solutions 1. Problems { Chapter 1 Problem 5.1. Show from rst principles that if V is a vector space (over R or C) then for any set Xthe space (5.1) F(X;V) = fu: X! Vg is a linear space over the same eld, with ‘pointwise operations’. Problem 5.2. If V is a vector space and SˆV is a subset which is closed
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