Suppose $T:\mathbb R^n \to\mathbb R^n$ is a linear transformation and suppose that $\mathbf v$ is a vector such that $T(\mathbf v) \ne 0$ but $T^2(\mathbf v) = 0$ (where $T^2 = T \circ T$) Prove that $\mathbf v$ and $T(\mathbf v)$ are linearlly independent.
I have no idea where to start with this! Any insight?
We must show that if $c_1 v + c_2 T(v) = 0$, then $c_1=c_2=0$.
Now apply $T$ to both sides of $c_1 v + c_2 T(v) = 0$ to obtain
$c_1T(v) + c_2 T^2(v) = T(0) = 0$. But $T^2(v) = 0$, so we get
$c_1T(v) = 0 \implies c_1 = 0$ since $T(v) \neq 0$. So $c_2T(v) = 0$, but $T(v) \neq 0 \implies c_2=0$ as well.