# Determine whether S is a subspace of the F-vector space V. In 2 cases.

For the following subsets $S$ determine whether $S$ is a subspace of the $F$-vector space $V$. Justify:

1. 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.

$$S = \{B\in V | AB \neq BA\}$$

1. $F$ = $\mathbb{C}$, $V$ = $\mathbb{C}$ and $S$ = $\mathbb{R}$, viewed as a subset of $\mathbb{C}$.
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For 1, $S$ cannot be subspace because the zero element in $V$, the zero $n\times n$ matrix $O$, does not belong to $S$, because $$AO=O=OA.$$
For 2, $S$ is not a subspace of the $F$-vector space $V$. To see this, let $1\in S=\mathbb{R}$, and $\sqrt{-1}\in F=\mathbb{C}$. However, $$\sqrt{-1}\cdot 1=\sqrt{-1}\not\in S=\mathbb{R}.$$
Note added: However, if $F=\mathbb{R}$, $V=\mathbb{C}$ and $S=\mathbb{R}$, then $S$ is a subspace of the $F$-vector space $V$