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.
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Hint: If $V$ has dimension $n$ and $v \in V$, then $v, Tv, T^2v, \dots, T^n v$ are linearly dependent.
lhf
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See also https://math.stackexchange.com/a/36662/589 – lhf Dec 29 '18 at 11:30
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Im not quite seeing it , can u elaborate pls? sorry – Someone Dec 29 '18 at 11:36
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@PedroSantos, there are $n+1$ vectors in that list. A linear dependence relation for them is the same as the action of a polynomial in $T$ on $v$. – lhf Dec 29 '18 at 11:41
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oh yeah right , thx – Someone Dec 29 '18 at 11:42