Prove that vectorspace is the result of a direct sum

Question

So I'm given the vectorspace $(\mathbb{R}, \mathbb{R}^{n \times n}, +)$ and the subspace $U$ which contains all symmetric matrices and W which contains all skew matrices. I'm asked to prove that the given vectorspace $\mathbb{R}^{n \times n} = U \bigoplus W$. How would I even start on this? I suppose I'll try to prove that every matrix is the sum of some symmetric and skew matrix?

Answer 1

Yes.

That is, you want to write arbitrary matrix $A$ as $A=B+C$ where $B=B^T$ and $C=-C^T$. What can you say about $A^T$= Then solve for $B$ and $C$.