Union of two vector subspaces not a subspace?
$U,W\subseteq V$ are subspaces.
Prove that in order for $U \cup W$ to be a subspace as well, either $U\subseteq W$ or $W\subseteq U$
Can anyone please give me a lead on this ?
Sure: supppose that $U$ is not contained in $W$ and $W$ is not contained in $U$. Let $x \in U \setminus W$, $y \in W \setminus U$, and consider $x+y$.
Note that this argument works for subgroups of an arbitrary group.
The condition that $U\not\subseteq W$ and $W\not\subseteq U$ is equivalent to the statement that there are $v_1\in U\setminus W$ and $v_2\in W\setminus U$, from which the result easily follows.