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Unformatted text preview: Problem Set 4 Math 115A/5 Fall 2011 Due: Friday, October 28 Problem 1 see (Theorem 2.10), (Exercise 2.3.8) Let V , W , X , and Y be vector spaces over a field F . Let P,Q L ( X,V ), T L ( V,W ), R,S L ( W,Y ). Prove the following: (a) T ( P + Q ) = TP + TQ and ( R + S ) T = RT + ST . (b) R ( TP ) = ( RT ) P . (c) TI V = T and I W T = T . (d) a ( RT ) = ( aR ) T = R ( aT ) for any a F . Problem 2 see (Theorem 2.12), (Exercise 2.3.5) Let n , m , p , and q be positive integers, and let F be a field. Let A,B M n p ( F ), C M m n ( F ), D,E M q m ( F ). Prove the following: (a) C ( A + B ) = CA + CB and ( D + E ) C = DC + EC . (b) D ( CA ) = ( DC ) A . (c) CI n = C and I m C = C . (d) a ( DC ) = ( aD ) C = D ( aC ) for any a F . Problem 3 This exercise should convince you that an inverse function does not always satisfy the same properties as the original function: Theorem 2.17 is not obvious! Let f : R R be the function given by f ( x ) = x 3...
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