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Question: 1 an nx n matrix is called symmetric if ata...

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1. An nx n matrix is called symmetric if AT-A. Let V I A E Maxn A is symmetric) (a) Show that V is a vector space, by showing that V is a subspace of M,xn. Use Lemma 4.3.2 on page 40 in the UC Davis text: show that (i) The additive identity belongs to V: Onxn E V, (ii) V is closed under addition, (ii) V is closed under scalar multiplication. (This method of showing V is a vector space is less tedious than verifying that V satisfies all of the axioms of a vector space.) (b) Determine a basis for V. Hint: Let Ey denote the matrix whose (i,j) and (j, i)-entries are1, with 0s elsewhere (c) What is the dimenson of V? (The dimension of a vector space V is the number of elements in a basis for V.)
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