Exercise 8.A.17

Proof. By the Fundamental Theorem of Linear Maps (3.22), we have

which implies that $\dim <!--\; FUNCTION\; APPLICATION\; -->\mathrm{null}<!--\; FUNCTION\; APPLICATION\; -->T^m=\dim <!--\; FUNCTION\; APPLICATION\; -->\mathrm{null}<!--\; FUNCTION\; APPLICATION\; -->T^{m+1}$ (because $\dim <!--\; FUNCTION\; APPLICATION\; -->\mathrm{range}<!--\; FUNCTION\; APPLICATION\; -->T^m=\dim <!--\; FUNCTION\; APPLICATION\; -->\mathrm{range}<!--\; FUNCTION\; APPLICATION\; -->T^{m+1}$). Thus, by 8.3, for all $k>m$, we have

Applying the Fundamental Theorem of Linear Maps again to $T^m$ and $T^{m+k}$ we see that

Since $\mathrm{range}<!--\; FUNCTION\; APPLICATION\; -->T^{m+k}\subset \mathrm{range}<!--\; FUNCTION\; APPLICATION\; -->T^m$, it follows that $\mathrm{range}<!--\; FUNCTION\; APPLICATION\; -->T^{m+k}=\mathrm{range}<!--\; FUNCTION\; APPLICATION\; -->T^m$. □