As the title says, I’m trying to show that if $E$ is a basis for subspace $U$ and $f+U, f\in F$ is a basis for the quotient space $V/U$, then the union $E\cup F$ is a basis for $V$.
I need to show independence and spanning. For spanning, take $v\in V$. Consider $v+U$. As $f+U, f\in F$ is a basis for $V/U$, we can write $v+U=a_1(f_1 + U) + \dots + a_r(f_r + U)$ for some scalars $a_i$.
How do I proceed?
Edit: I have already proved linear indpendence.