I have two questions regarding subspaces of vector spaces. They are exercises for a course, and are in that sense not homework, since I will not get assessed on it. Some good hints are what I am looking for, rather than complete answers.
Question 1: Let $V',V'',W$ be subspaces of $V$. Does the inclusion $(V' + V'') \cap W \subseteq (V' \cap W) + (V'' \cap W)$ always hold? I have not been able to find a counter-example with simple subspaces of $\mathbb{R}^3$, and have not come up with a proof. My intuition is that the inclusion does not hold.
Question 2: Let $V',V''$ be subspaces of $V$. Show that if $V' \cup V''$ is a subspace of $V$, then either $V' \subseteq V''$ or $V'' \subseteq V'$.
I guess one proof strategy is to assume that $V' \nsubseteq V''$ and then show that $V'' \subseteq V'$. Another would be to assume $V' \nsubseteq V''$ and $V'' \nsubseteq V'$ and then show that $V' \cup V''$ is not a subspace.