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I'm troubled with the following question:

$V_1$ and $V_2$ are subspaces of $K$-vector space $V$. Show that $V_1 \cup V_2 = V$ implies $V_1 = V$ or $V_2 = V$.

I don't understand why $V_1$ or $V_2$ is supposed to equal $V$ when $V_1 \cup V_2 = V$ shows that they are subsets.

Jan
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endrasa
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1 Answers1

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If $V_1\subset V_2$ or $V_2\subset V_1$ then the statement is obvious. If not, then there exixts $v_1\in V_1 \setminus V_2$ and $v_2\in V_2\setminus V_1$. It is easy to check, that $v_1+v_2\not\in V_1\cup V_2$.

Boris Bilich
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