I want to prove that if $W_1, W_2$ are linear subspaces of $V$, then $W_1 \cup W_2$ is a linear subspace of $V$ if and only if $W_1 \subset W_2$ or $W_2 \subset W_1$. I sort of "see" it right away. The problem here is formalization. In particular, how do I show that if I take two elements, each in only one subspace, their sum need not be in $W_1 \cup W_2$?
Suppose $u\in W_1\setminus W_2$ and $v\in W_2\setminus W_1$. If $u+v\in W_1\cup W_2$, then by definition either $u+v\in W_1$ or $u+v\in W_2$. However, we have:
- If $u+v\in W_1$, since $u\in W_1$ and $W_1$ is a subspace, we also have $v=(u+v)-u\in W_1$, contradicting the choice of $u,v$.
- Similarly, $u+v\in W_2$ would imply $u\in W_2$, which is absurd.
Thus, either $W_1\setminus W_2$ or $W_2\setminus W_1$ is empty, which implies either $W_1\subseteq W_2$ or $W_2\subseteq W_1$.