I come across an interesting problem on my journey of cracking open some old math books and cracking down on problems from boredom. I cannot seem to wrap my head around this problem of subspaces. The problem is,
Let $W_1$ and $W_2$ be subspaces of a finite dimensional vector space $V$. Can the following be proved?
(a) $W_1+W_2=\{w_1+w_2:w_1 \in W_1,w_2 \in W_2\}$ is a subspace of $V$.
(b) $W_1 \cap W_2$ is a subspace of $V$.
(c) $\dim(W_1)+ \dim(W_2)= \dim(W_1+W_2)+ \dim(W_1 \cap W_2)$.
Any ideas on how to go about solving this?
Thank you in advance.