Assume the notation of Theorem 6.32. (a) Prove that for any ordered basis 0, ipp is
Chapter 6, Problem 8(choose chapter or problem)
Assume the notation of Theorem 6.32. (a) Prove that for any ordered basis 0, ipp is linear. (b) Let 0 be an ordered basis for an n-dimensional space V over F, and let 4>p: V * Fn be the standard representation of V with respect to 0. For A G MnXn (F), define 77: V x V -> F by H(x,y) = [(f>0(x)\tA[(j)fi{y)]. Prove that 77 G #(V). Can you establish this as a corollary to Exercise 7? (c) Prove the converse of (b): Let 77 be a bilinear form on V. If A = MH), then H(x,y) = [M^Y^Mv)]-
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