Let $\alpha: U \rightarrow V$ and $\beta: V \rightarrow W$ be linear maps between finite-dimensional real vector spaces.

Show that the rank $r(\beta \alpha)$ satisfies $r(\beta \alpha) \leqslant \min (r(\beta), r(\alpha))$. Show also that $r(\beta \alpha) \geqslant r(\alpha)+r(\beta)-\operatorname{dim} V$. For each of these two inequalities, give examples to show that we may or may not have equality.

Now let $V$ have dimension $2 n$ and let $\alpha: V \rightarrow V$ be a linear map of rank $2 n-2$ such that $\alpha^{n}=0$. Find the rank of $\alpha^{k}$ for each $1 \leqslant k \leqslant n-1$.