Exercise 10.A.17

Proof. Let $u_1,\dots ,u_m$ be a basis of $\mathrm{null}<!--\; FUNCTION\; APPLICATION\; -->T$. Extend it to a basis $u_1,\dots ,u_m,v_1,\dots ,v_n$ of $V$. Then $T{v}_1,\dots ,T{v}_n$ is a basis of $\mathrm{range}<!--\; FUNCTION\; APPLICATION\; -->T$ (see the proof of 3.22). Define a linear map $S^{\prime }:\mathrm{range}<!--\; FUNCTION\; APPLICATION\; -->T\to V$ by

for $j=1,\dots ,n$. We can extend $S^{\prime }$ to an operator $S$ on $V$. Regardless of how we extend $S^{\prime }$, it is clear that the matrix of $\mathit{ST}$ will be of the following form

where the $1$’s appear on the last $n$ diagonal entries. However, the trace of this matrix should equal $0$ (since $\mathrm{trace}<!--\; FUNCTION\; APPLICATION\; -->(\mathit{ST})=0$). Hence $n=0$, because the $1$’s should not appear in any of the columns. Therefore $T=0$. □