Answer to Question #12467 in Linear Algebra for john.george.milnor
Prove that any matrix ring as vector space is direct sum of sets of symmetric and antisymmetric matrices.
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 + AT ) is symmetric and A2 = 1/2(A - AT ). Moreover, A = A1 + A2. And that's it.