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Let $F$ and $G$ be subspaces of a vector space $E$. When is $F \cup G$ a subspace ?

When is $F \cap G$ a subspace?

Thank you in advance.

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I think in the second case always. However, I don't know about the first case. – Carpediem Feb 9 '13 at 21:32
Okay, what makes you think the second case will always be a subspace? And in the first case, can you think of counterexamples for when it is not a subspace? Often knowing a few counterexamples can go a long way into providing intuition of when or why something might be true/false. – Clayton Feb 9 '13 at 21:34

Hint: Intersection keeps only those things that are in both subspaces. So if two vectors are in the intersection, then both are in each subspace, thus their sum is in each subspace, thus their sum is in the intersection. Now, formalize this and complete the proof that indeed the intersection of any two subspaces is a subspace.

As for 1), first build some intuition by considering example. Take $\mathbb R^2$, a nice vector space where you can visualize things easily (or just draw things on a piece of paper). So, now take the $X$ axis and the $Y$ axis. These are subspaces. Does their union look linear? Can you find two vectors in the union whose sum is not in the union.

For a more formal proof start with two subsapces $F,G\subseteq E$ and assume their union is a subspace. Can you have a situation in which you have $v\in F$, $w\in G$ but $w\notin G$ and $v\notin G$?

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+1: Good answer and it still leaves some of the details to the student. – Clayton Feb 9 '13 at 21:39
+1, very good. – Rustyn Feb 9 '13 at 21:41

For a partial answer:
The union of two subspaces is a subspace if and only if one of the subspaces is contained in the other.

See this Related Question for more information; Gerry Myerson discusses this in detail.

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