Prove or disprove If $\{u_1, u_2, u_3,u_4,u_5 \}$ form basis in $\mathbb{R}^5 $ and $V$ is 2-d subspace then show that V has basis from $\mathbb{R}^5$

Question

I have a doubt. If $\{u_1, u_2, u_3,u_4,u_5 \}$ form basis in $\mathbb{R}^5 $ and $V$ is 2-dimensional subspace of $\mathbb{R}^5$ then show that V has basis of 2 elements from given basis of $\mathbb{R}^5$

Is this correct?

Answer 1

As others have said, this statement is false in general. I will give you this counter-example:

If $u_1=(1,0,0,0,0),\; u_2=(0,1,0,0,0),\; u_3=(0,0,1,0,0),\;\ldots$ and $V=\{(a,a,b,b,b):a,b\in\mathbb{R} \}$. It is clear that for every member of the basis: $u_i \notin V$. Therefore, it is impossible for any pair of the original basis vectors to span $V$.