Q:

Supplementary subspaces of space of square matrices

Let $M_n(K)$ be the vector space of square matrices of size n, and
$$M_n^s(k) = \{A \in M_n(K) | A^t = A\}$$
and
$$M_n^a(k) = \{A \in M_n(K) | A^t = -A\}$$
subspaces.
I need to show that these subspaces are supplementary in $M_n(K)$.
I guess I should show that $M_n^s(k) + M_n^a(k) = M_n(K)$ and that $M_n^s(k) \cap M_n^a(k) = \{0\}$, but I don't know how to do this. Or is there another way?

A:

Hint.
show that: $\forall A \in M_n(K)$ there is a decomposition as sum of a symmetric ant an anti-symmetric part as
$$
A=A_s+A_a 
$$
with 
$$A_s=\frac {A+A^T}{2} \qquad A_a=\frac {A-A^T}{2}
$$

