Proving that matrix to the power multiplied by a non-zero vector are independent from same matrices to different powers

Question

Suppose I have a $n$x$n$ square matrix $B$ and $n$-length column vector $a \ne \underline{0}$, such that

\begin{align*} B^{3}a &= \underline{0} \\ B^{2}a &\ne \underline{0} \end{align*}

I know this must mean that \begin{equation} \nexists c\in\mathbb{R}, B^{3}=cB^{2} \end{equation}

Can I therefore prove that all $B^{n}a$ are independent?

Answer 1

We could have a situation where $B^3 = 0$ yet $B^2 \neq 0$. For instance, take $B =\begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0& 0 \end{pmatrix}$ and $a = e_3$. Then $c = 0$ satisfies the equation $B^3 = cB^2$.