Let $V$ be a vector space and $U,W$ be its subspaces. Prove that if the union $U\cup W$ is a subspace of $V$, then $W \subseteq U$ or $U \subseteq W$

Question

Let $V$ be a vector space and $U,W$ be its subspaces. Prove that if the union $U\cup W$ is a subspace of $V$, then $W \subseteq U$ or $U \subseteq W$.

I'm not sure where to begin at all really.

Answer 1

Begin by removing the unnecessary distractions: if $H,K$ are subgroups of a group $(G,\cdot)$ and if $H \cup K$ is also a subgroup of $G$, then $H\subset K$ or $K\subset H$

Proof: suppose $H \not \subset K$. There exists $h \in H -K$. Let $k \in K$. Then $hk \in H \cup K$ but not in $K$, because otherwise $h \in K$. So that $hk \in H$, but then $k \in H$.

Answer 2

Suppose that $U\cup W$ is a subspace.
If $U\subset W$, we are done.
If $U\not\subset W$, there exists $u\in U$ such that $u\notin W$.
Let $w\in W$, we have $w+u\in U \cup W$.
If $w+u\in W$, then $u=(w+u)-w\in W$, a contradiction.
Hence, $w+u \in U$ and thus $w = (w+u)-u \in U$.
It follows that $W\subset U$.

Thanks to Luca Bressan, the proof can be simplified as follow:

Suppose that $U\cup W$ is a subspace.
Let $u\in U$ and $w\in W$, then $w+u\in U \cup W$.
If $w+u\in W$, then $u=(w+u)-w\in W$.
If $w+u \in U$ and thus $w = (w+u)-u \in U$.
It follows that either $W\subset U$ or $U\subset W$.