Let V = Fn , and let A E M.X/I(/"). (a) Prove that (x. Ay) = (A*x,y) for all x,y C V
Chapter 6, Problem 23(choose chapter or problem)
Let V = Fn , and let A E M.X/I(/"). (a) Prove that (x. Ay) = (A*x,y) for all x,y C V. (b) Suppose that for some B Mnxn (F), we have (x,Ay) = (Bx,y) for all x.y E_ V. Prove that B = A*. (c) Let a be the standard ordered basis for V. For any orthonormal basis 0 for V, let Q be the n x n matrix whose columns are the vectors in ft. Prove that. Q* =Q '. (d) Define linear operators T and U on V by T(x) Ax and U(x) = A*x. Show that [\J]p [T]g for any orthonormal basis ft for V
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