Exercise 8.A.10

Proof. If $T$ is not nilpotent, then $\dim <!--\; FUNCTION\; APPLICATION\; -->\mathrm{null}<!--\; FUNCTION\; APPLICATION\; -->T^n<n$ and , by the same reasoning used in 8.4, it follows that $\mathrm{null}<!--\; FUNCTION\; APPLICATION\; -->T^{n-1}=\mathrm{null}<!--\; FUNCTION\; APPLICATION\; -->T^n$. Thus, by 8.5, we have

Since $\mathrm{range}<!--\; FUNCTION\; APPLICATION\; -->T^n\subset \mathrm{range}<!--\; FUNCTION\; APPLICATION\; -->T^{n-1}$, we must also have

Then, by the Fundamental Theorem of Linear Maps (3.22),

3.78 now implies that $\mathrm{null}<!--\; FUNCTION\; APPLICATION\; -->T^{n-1}+\mathrm{range}<!--\; FUNCTION\; APPLICATION\; -->T^{n-1}$ is a direct sum. □