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Let $K=\Bbb F_2$ and $V$ be a vector space over K.
Let $A,B,C$ be subspaces of $V$.

If $A\cup B\cup C$ is a subspace of $V$, then one of $A,B,C$ is contained in the union of the other two, ie one of the following holds: $$A \subseteq B\cup C$$ $$B \subseteq C\cup A$$ $$C \subseteq A\cup B$$

How do I show if this is right or wrong?

cineel
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Mathisnothealthyforu
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  • Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. – Community Jan 02 '22 at 18:01

1 Answers1

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Take $V = K^2.$ Then pick $A = \{(0, 0), (1, 0)\}, B = \{(0, 0), (0, 1)\},$ and $C = \{(0, 0), (1,1)\}.$ Thus $A \cup B \cup C = V$ is a subspace, but none of your inclusions hold.

  • Thank you! Simple but clever. Is the answer still the same if 1+1≠0 applies in K? – Mathisnothealthyforu Jan 02 '22 at 18:06
  • @Mathisnothealthyforu The theorem is true in characteristic 0. –  Jan 03 '22 at 17:02
  • @Mathisnothealthyforu More generally, looking at my argument, it's true if $K$ is any infinite field. I believe that with some work one could probably find a way to construct a counterexample to the finite field case. –  Jan 03 '22 at 17:09
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    @Michael i think it's true if char K $\neq$ 2. – cineel Jan 04 '22 at 16:03
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    @cineel Ah, actually I think it's true for any field except $\mathbb{Z}/2\mathbb{Z}.$ Take a field with three or more distinct elements, say $0, 1, x.$ Suppose you have a counterexample to the theorem. Pick $a\in A, a\not\in B\cup C$ and similarly choose $b.$ Then $a+0b, a+1b, a+x b$ all lie in $A\cup B\cup C$ since that union is a subspace. But none of these lie in $B$ (since $a\not\in B$), and $a+b, a+xb$ cannot lie in $A$ (since $b\not\in A$). Thus $a+b, a+xb$ in $C,$ but $x\neq 1$ then implies $(x-1)b\in C$ so that $b\in C,$ a contradiction. Thus this only fails over $\mathbb{F}_2.$ –  Jan 04 '22 at 18:29
  • This was essentially the same argument I knew for infinite fields, but I didn't realize I could tweak it slightly to only need 3 scalars instead of infinitely many. –  Jan 04 '22 at 18:29