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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.

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I think I've seen those questions asked here already. – tomasz Aug 6 '12 at 15:16
up vote 2 down vote accepted
  1. Do $V´=(x,0,0),V´´=(0,x,0)$ and $W=(x,x,0)$ where $x \in \mathbb{R}$.
  2. Let $x$ be such that $x\in V,´x \notin V´´$ and $y\in V´´,y \notin V´$. Then if $V´\cup V´´$ is a subspace we have $1/2x+1/2y \in V´\cup V´´$, but $$1/2x+1/2y \in V´ \Rightarrow (1/2x+1/2y) - 1/2x = 1/2 y \in V'\Rightarrow y \in V'$$ and $$ 1/2x+1/2y V´´ \Rightarrow (1/2x+1/2y) - 1/2y = 1/2 x \in V'´\Rightarrow x \in V´´.$$ Hemce $\dfrac{1}{2}x +\dfrac{1}{2}y \notin V´+ V´´$ contradiction.
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For question #1: Examine $\mathbb{F}\times \mathbb{F}$, and let $V'=\mathbb{F}\times\{0\}$, $V''=\{0\}\times\mathbb{F}$, and let $W=\{(b,b)\mid b \in \mathbb{F}\}$.

For question #2: Probably the best way is to assume that $V'\subsetneq V''$ and $V''\subsetneq V'$, so that you are able to pick elements $x$ and $y$ so that $x\in V'\setminus V''$, and $y\in V''\setminus V'$. Ask yourself where $x+y$ would fall.

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