Problem 3. Let Maxn(R) denote the vector space of all n x n matrices with real entries. Let S be the subset of all symmetric matrices, i.e., S = {X € Mnxn (R) : X¹ = X}, and let A be the set of all anti-symmetric matrices, i.e., A = {X € Mnxn (R) : X¹ = -X}. We proved in Homework 2 that S and A are subspaces of Mnxn (R), and that Mnxn(R) SO A. = (a) Find bases for S and A. Note: You need to prove that your examples are in fact bases. (b) Use this information to compute the dimensions of S, A, and Mnxn (R).

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