Question

Prove that any matrix ring as vector space is direct sum of sets of symmetric and antisymmetric matrices.

Answer 1

It is evident, that it is sufficiently to prove that any matrix A can be written as A1 + A2, where A1 is symmetric and A2 is antisymmetric matrices.
One can easily see that A1 = 1/2(A + A^T ) is symmetric and A2 = 1/2(A - A^T ). Moreover, A = A1 + A2. And that's it.