Chapter 4.3, Problem 31P

Problems

Let $S_1$ and $S_2$ be subspaces of vector space $V$. Let

$S_1\cup S_2=\left\{v\in V:S_1\text{or }v\in S_2\right\}$.

$S_1\cap S_2=\left\{v\in V:S_1\text{and }v\in S_2\right\}$.

and let

$S_1+S_2=\left\{v\in x+y\text{for some }x\in S_1\text{and }y\in S_2\right\}$.

(a) Show that, in general $S_1\cup S_2$ is not a subspace of $V$.

(b) Show that $S_1\cap S_2$ is a subspace of $V$.

(c) Show that $S_1+S_2$ is a subspace of $V$.