Chapter 4.3, Problem 58E

Proof Let $V$ and $W$ be two subspaces of vector space $U$.

$\left(\text{a}\right)$ Prove that the set

$V+W=\left\{u:u=v+w\text{,}v\in V\text{ and }w\in W\right\}$

is a subspace of $U$.

$\left(\text{b}\right)$ Describe $V+W$ when $V$ and $W$ are subspaces of $U=R^2:$

$V=\left\{\left(x,0\right):x\text{is}\text{a}\text{real}\text{number}\right\}$ and $W=\left\{\left(0,y\right):y\text{is}\text{a}\text{real}\text{number}\right\}$.