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A matrix A is symmetric if AT = A and skew-symmetric if AT = -A. Let...

A matrix A is symmetric if AT = A and skew-symmetric if AT = -A. Let Wsym be the set of all symmetric matrices and let Wskew be the set of all skew-symmetric matrices

(a) Prove that Wsym is a subspace of Fn×n . Give a basis for Wsym and determine its dimension.

(b) Prove that Wskew is a subspace of Fn×n . Give a basis for Wskew and determine its dimension.

(c) Prove that F n×n = Wsym ⊕Wskew. Verify that dim(Fn×n ) = dim(Wsym)+ dim(Wskew)

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