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Problem: Let $V$ be a finite dimensional vector space and $V_1,\ldots,V_n\subset V$ vector subspaces. Show that if $W\subset V$ is a vector subspace and $$W\subset V_1\cup\cdots\cup V_n,$$ then $$W\subset V_k$$ for some $1\le k\le n$.

I get the intuitive idea of why this should hold by drawing pictures, but I can't prove it rigorously. I know that if $V_1,V_2$ are two subspaces and $V_1\cup V_2$ is also a subspace, then $V_1\subset V_2$ or $V_2\subset V_1$. Does that help?

user349918
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  • I think you must be making an assumption about the underlying field; this isn't true for finite fields. Are you thinking of vector spaces over $\mathbb R$? – joriki Jun 24 '16 at 10:51
  • @joriki $\Bbb R$ or $\Bbb C$. – user349918 Jun 24 '16 at 10:52
  • @joriki: It's true for finite fields under an assumption on the cardinality of the field. See my answer to this question. – Bernard Jun 24 '16 at 11:23
  • @Bernard: Well, that assumption depends on $n$; I meant that it isn't true for general $n$ as stated, and indeed there's no finite field for which it's true for all $n$. – joriki Jun 24 '16 at 11:50

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