Assume that $W$ and $V$ are two subspace of $X$. Is their union a subspace of $X$ too? I think it is not true unless under certain conditions but I do not know what conditions...
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What conditions have you tried? – tmastny Oct 20 '14 at 14:53
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1Hint for you to prove: $;W\cup V;$ is a subspace of $;X;$ iff $;W\subset V;$ or $;V\subset W;$ . – Timbuc Oct 20 '14 at 15:02
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1(A search for "union of two subspaces" produces at least six questions on this topic, with many answers.) – MJD Oct 20 '14 at 15:12
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A necessary and sufficient condition is that
$$W\subseteq V \text{ or } V\subseteq W.$$The sufficiency is clear.
If $W\bigcup V$ is a subspace, suppose that there exist an $x\in W-V$, and a $y\in V-W$. Then we have $x+y\notin W\bigcup V$, contradiction! Hence, either $x$ or $y$ does not exist, which leads to the desired conclusion.
Eclipse Sun
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No, union is not vector subspace in general (e.g. $\{y=x\}\cup \{y=-x\}$ in $\mathbb{R^2}$... direct sum $W\oplus V$ is a vector subspace).
Milly
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