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Let V be a vector space, W and W' be linear subspaces of V. Show that:

$W\cup W'$ is a linear subspace is equivalent to $W\subseteq W'$ or $W'\subseteq W$

I have already shown that left follows from right (<-), but I can't figure out the conclusion from left to right (->).

Joey
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1 Answers1

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If we suppose that $W \not \subset W'$ and $W' \not \subset W$, then we can find a vector $v \in W$ that is not in $W'$ and a vector $v' \in W'$ that is not in $W$.

What can we say about the vector $v + v'$?

Joel
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  • It's a combination of elements of $W\cup W'$ that is not in $W\cup W'$ itself. Ergo, $W\cup W'$ is not a linear vector space. Proof by contradiction! Thanks a lot! – Joey Nov 06 '13 at 23:54
  • Whats the argument for v+v' not being an element of the unity though? – Joey Nov 07 '13 at 00:04
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    @Joey Just think a bit about in which subspace it may be (the difference of 2 vectors in a subspace is still in that subspace) – fedja Nov 07 '13 at 02:24
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    You guys are awesome. This is my first time asking a question. I must say it probably won't be the last. I like your ways of helping without giving me the whole solution. – Joey Nov 07 '13 at 07:11