Prove that $v, Tv, T^2v, ... , T^{m-1}v$ is linearly independent
It is simple to do it directly:
If $a_0 v + a_1 Tv +a _2 T^2v + \cdots + a_{m-1} T^{m-1}v=0$, then apply $T^{m-1}$ and get $a_0 T^{m-1}v=0$, which implies $a_0=0$. Now apply $T^{m-2}$ to get $a_1=0$. And so on.
The matrix $T$ need not be nilpotent. Let's take a simple example: $n=2$. Say you have a vector $v$ with $Tv\ne0$ but $T^2v=0$. Let $w=Tv$. We need to show that $av+bw=0$ implies $a=b=0$. But applying $T$ gives $$0=T(av+bw)=aTv+bTw=aw.$$ As $w\ne0$, $a=0$. Therefore $bw=0$ and so $b=0$.
Can you do something similar for $n=3$? General $n$?