$U \cup W$ not a vector subspace.

Question

Let $U, W$ be two vector subspace in a vector space $V$. Assume that neither one is contained in the other, i.e. $U \not\subset W$ and $W \not\subset U$. What is the easiest way to see that $U \cup W$ is not a vector subspace?

Answer 1

Hint:

Consider a vector $u\in U\smallsetminus W$ and a vector $w\in W\smallsetminus U$. Where does $u+w$ live?