If $T^rv=v$, are $v,Tv,\ldots,T^{r-1}v$ linearly independent?

Question

Let $V$ be a finite-dimensional vector space over the field of complex numbers, and let $T:V \to V$ be a linear map.

Suppose that we have a nonzero vector $v \in V$ such that $T^rv=v$, and that $r$ is minimal, i.e. $T^kv \neq v$ for $0<k<r$.

Are $v,Tv,\ldots,T^{r-1}v$ linearly independent?

I tried to assume dependence and apply various powers of $T$, but so far came up with nothing...

Answer 1

Not true. Suppose $Tv=-v$ with $v \neq 0$. Then $T^{2}v=v$ and $Tv \neq v$. But $v,Tv$ are not independent.