Let $m \geq 2$ be a natural integer and $E$ an $\mathbb{R}$-vector space of dimension $2m+1$. Let $T$ be an endomorphism of $E$ satisfying (H1): $T^{2m} \neq 0_{\mathcal{L}(E)}$ and $T^{2m+1} = 0_{\mathcal{L}(E)}$.
Deduce that for all $k \in \{0, \ldots, 2m+1\}$, we have
$$\operatorname{dim}(\operatorname{Im}(T^k)) = 2m+1-k, \quad \operatorname{dim}(\operatorname{ker}(T^k)) = k$$