Union of two vector subspaces not a subspace? I'm having a difficult time understanding this statement. Can someone please explain with a concrete example?
The reason why this can happen is that all vector spaces, and hence subspaces too, must be closed under addition (and scalar multiplication). The union of two subspaces takes all the elements already in those spaces, and nothing more. In the union of subspaces $W_1$ and $W_2$, there are new combinations of vectors we can add together that we couldn't before, like $v_1 + w_2$ where $v_1 \in W_1$ and $w_2 \in W_2$. 

For example, take $W_1$ to be the $x$-axis and $W_2$ the $y$-axis, both subspaces of $\mathbb{R}^2$.
Their union includes both $(3,0)$ and $(0,5)$, whose sum, $(3,5)$, is not in the union. Hence, the union is not a vector space.
To prove that a vector(U) is a subspace of a vector space(V).
we need to prove that a+$\alpha $b,(where $\alpha$ is any scalar belonging to the field of the vector space) belongs to U.

so we will make that the two vectors make a single vector. And that is possible only when one of the vector space is a subset of the other.