Vector space of finite dimension then $Tor_{K[x]}V=V$

Question

Alright guys, I have a doubt. Let $K$ be a field and $V$ a vector space of dimesion $n$. Because of this we know $V$ is a finitely generated free module. The goal is to show that $Tor_{K[x]}V=V$ where $V$ has a $K[x]$ module structure induced by $T \in End_k(V)$, $x\cdot v=T(v)$. So I tried working with the generators and try to show that there is only a finite number of options where they can be send to by the endomorphism to prove that indeed for all $v\in V$ there is a polinomyal that is going to kill it, but I'm not very confident about it. Can you help me out? Thanks.

Answer 1

Hint: If $V$ has dimension $n$ and $v \in V$, then $v, Tv, T^2v, \dots, T^n v$ are linearly dependent.