Let $U$ be the subspace {$(x,y,z)∈ \Bbb R^3:x=0 , y=0$} of $\Bbb R^3$

Question

Let $U$ be the subspace {$(x,y,z)∈ \Bbb R^3:x=0 , y=0$} of $\Bbb R^3$. Then show that if $ v_1,v_2∈R^3$ be vectors such that the set {$v_1+U,v_2+U$} span {$\Bbb R^3/U$}. Then {$v_1,v_2,v_3$} doesn't span $\Bbb R^3$ for any $v_3∈ \Bbb R^3$

So when I tried solving the question as follows

{$v_1+U,v_2+U$} spans the space {$\Bbb R^3/U$} $\implies$ $a(v_1 + U) + b(v_2 + U) = v + U $ for any $v$ in $\Bbb R^3$ $\implies$ $av_1 + bv_2 - v \in U$

From here if we can show that $v_1,v_2$ and $v$ are linearly dependent we would have solved the question. But I'm unable to proceed and would like help, with this method or any other. Thankyou.

Answer 1

You cannot prove it, since it is false. Simply take $v_1=(1,0,0)$, $v_2=(0,1,0)$, and $v_3=(0,0,1)$. Then $\{v_1+U,v_2+U\}$ is a basis of $\Bbb R^3/U$ and $\{v_1,v_2,v_3\}$ spans $\Bbb R^3$.