TITLE: Linear Algebra - basis question
QUESTION [6 upvotes]: 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.

REPLY [11 votes]: 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 $\{1,x,x^2+x^3\}$.