Given the linear transformation from $S: M_{n}(\mathbb{R}) \to M_{n}(\mathbb{R})$ defined by $S(A) = A + A^{T}$, where $A$ is a fixed $n \times n$ matrix, how do you find the $\dim(\ker(S))$?
I've found that $\ker(S) = \mbox{the set of all matrices whose transpose, negated, is itself}$, but how can I find the dimension? I'm having difficulty coming up with a basis.
Thanks in advance.