Consider $W\subseteq V$, a subspace over a field $\mathbb{F}$ and $T:V\rightarrow V$ a linear transformation with the stipulation that $T(W)\subseteq W$. Then we have the induced linear transformation $\overline{T}:V/W \rightarrow V/W$ such that $\overline{T}=T(v)+W$.
I'm supposed to show that this induced transformation is well-defined, and that given $V$ finite and $T$ an isomorphism that $\overline{T}$ is an isomorphism. I'm having a little trouble with this part. Namely, I want to show that the $ker(\overline{T})=W$, and I'm to the point where I realize that this means $T(v)\in W$. How do I know there's not some random $v\in V\setminus W $ such that $ T(v)\in W$.
In particular, is all this true if $V$ is not finite dimensional? I can't immediately think of a counterexample...