\textbf{Cyclic subspaces}\\
Let $r$ be a nonzero natural integer. Prove that the smallest vector subspace $\mathcal{D}_r$ of $\mathcal{D}$ containing $X^{-r}$ and stable by $\xi$ admits as basis $(X^{k-r})_{0 \leqslant k \leqslant r-1}$. Write the matrix of the endomorphism $\xi_{\mathcal{D}_r}$ induced by $\xi$ on $\mathcal{D}_r$ in this basis.