Let $\mathbb{F}$ be a field and $A\in M_n(\mathbb{F})$. Let the minimal polynomial of $A$ be $\prod_{i=1}^m q_i(x)^{\ell_i}$, where each $q_i(x) \in \mathbb{F}[x]$ is monic irreducible, each $\ell_i>0$, and gcd$(q_i(x),q_j(x))=1$ for $i \neq j$. If $s=\deg q_1(x)^{\ell_1}$, does there exist a $v \in \mathbb{F}^n$ such that $\lbrace v, Av, ..., A^{s-1}v \rbrace$ is linearly independent?
I am aware that the above is true when $m=1$. But what if $m>1$?