Let $V$ be a vector space over $\mathbb{F}$ and $h : V \rightarrow V$ a linear map. For $p \in \mathbb{F}[X]$ and $v \in V$ we define a multiplication $p \cdot v = p(h)(v)$. Now let's say $$h(v) = \begin{pmatrix} 5 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 5 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 5 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 5 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 3 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 \end{pmatrix}v.$$
Find a decomposition of $V$ as direct sum of five $\mathbb{F}[X]$ modules, that can be generated by a single vector and explicitly specify these vectors.
Any advice is greatly appreciated.