Let $V$ be a finite dimensional vector space. Let $U$ be a subspace of $V$. Then $V/U=\{v+U:v\in V\}$.

Addition on $V/U$ is defined by $(v+U)+(w+U)=(v+w)+U$ and scalar multiplication is defined by $\lambda (v+U)=\lambda v+U$.

If a basis of $V$ is $v\_1,\dots,v\_n$. Is $v\_1+U,\dots,v\_n+U$ a basis of $V/U$ using the operations of addition and scalar multiplication above? For every $v\in V$ we have that $v=a\_1v\_1+\dots+a\_nv\_n$. Then using the definition of addition and scalar multiplication defined above, for $v+U\in V/U$ we have that $v+U=a\_1(v\_1+U)+\dots+a\_n(v\_n+U)$.

The reason I ask this question is that, if this is true, then that would imply that $\dim V=\dim V/U$. But another result tells us that $\dim V/U=\dim V-\dim U$.

If the above result were to hold, that would imply that $\dim U=0$ and that $U=\{0\}$. Which doesn't have to the case always. I am pretty sure I am missing something, but I just can't figure out what.