Exercise 8.A.15

Proof. By the same reasoning used in the proof of 8.4, it follows that $\dim <!--\; FUNCTION\; APPLICATION\; -->\mathrm{null}<!--\; FUNCTION\; APPLICATION\; -->N^{\dim <!--\; FUNCTION\; APPLICATION\; -->V}\ge \dim <!--\; FUNCTION\; APPLICATION\; -->V$. Thus $\dim <!--\; FUNCTION\; APPLICATION\; -->\mathrm{null}<!--\; FUNCTION\; APPLICATION\; -->N^{\dim <!--\; FUNCTION\; APPLICATION\; -->V}=\dim <!--\; FUNCTION\; APPLICATION\; -->V$ and so $N$ is nilpotent. We have $\dim <!--\; FUNCTION\; APPLICATION\; -->V+1$ null spaces each of different dimension. Since the sequence

must be sorted in strictly increasing order, the only way this can fit is if $\dim <!--\; FUNCTION\; APPLICATION\; -->\mathrm{null}<!--\; FUNCTION\; APPLICATION\; -->N^j=j$ for each $j$. □