Let V be a vector space of nxn Matrices over a field $\displaystyle K$. Show that the following subsets $\displaystyle U$ and $\displaystyle W$ are subspaces of $\displaystyle V$.

1. $\displaystyle U = {A = (a_ij) \in V | a_ij = a_ji, \forall 1 \leq i, j \leq n}$.

2. Let $\displaystyle T$ be a fixed matrix in $\displaystyle V$ and $\displaystyle W = {A \in V | AT = TA}$.

So, $\displaystyle A$ are symmetrical matrices in $\displaystyle \mathbb{R}^n$, is there anything special about symmetrical matrices relevant for this question (part1)?

If the matrix is symmetrical than (part2) $\displaystyle AT \equiv TA$?