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Symmetric Matrices as a Subspace of all Square Matrices

Theorem: subspace W of a vector space V over R is a subset of V which also has the properties that W is closed under addition and scalar multiplication. That is, For all x, y in W, x and y are in V and for any c in R, cx + y is in W.
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Let W_n be the subset of all M_(n x n)(R) such that A_ij = A_ji (ie. the set of all real symmetric square matrices).