I am revising for a Linear Algebra exam by going through some previous quiz questions, that I have True/False answers to, but not the reasoning or counterexamples. I am stuck on the following:

If $v_1,v_2,v_3,v_4$ is a basis for $V$, and $U$ is a subspace of $V$ such that $v_1,v_2\in U$ but $v_3,v_4\notin U$, then $v_1,v_2$ is a basis of U.

The answer is listed as False, which intuitively seems right, but I can't seem to find a counterexample.

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    $\begingroup$ Hint: look for a three dimensional subspace of $V$ that contains $\vec v_1,\vec v_2$ but neither of the others. $\endgroup$ – lulu Aug 5 '17 at 18:11

Consider the set $V$ of polynomials in $\Bbb R[X]$ with degree $\le 3$.

Then $\{1,x,x^2,x^3\}$ is a basis for $V$. Let $U$ be the subspace generated by $\{1,x,x^2+x^3\}$.

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    $\begingroup$ To expand on this answer, let $U$ be the subspace generated by $v_1$, $v_2$, and ANY non-trivial linear combination of $v_3$ and $v_4$. $\endgroup$ – Matthew Aug 6 '17 at 3:51

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