For the following subsets $S$ determine whether $S$ is a subspace of the $F$-vector space $V$. Justify:
- Let $F$ = $\mathbb{R}$ and $V$ = $M_n(\mathbb{R})$, the set of $n \times n$ matrices with real entries. Let $A$ be a particular $n \times n$ matrix with real entries.
$S = \{B\in V | AB \neq BA\}$
- $F$ = $\mathbb{C}$, $V$ = $\mathbb{C}$ and $S$ = $\mathbb{R}$, viewed as a subset of $\mathbb{C}$.