Exercise 10.A.8

Proof. If $w=0$, then $T=0$ and $\mathrm{trace}<!--\; FUNCTION\; APPLICATION\; -->T=0$. If $w\ne 0$, then extend $w$ to a basis of $V$. All entries of the matrix of $T$ with respect to this basis are $0$, except possibly the first row, because $\mathrm{range}<!--\; FUNCTION\; APPLICATION\; -->T\subset \mathrm{span}<!--\; FUNCTION\; APPLICATION\; -->(w)$. Therefore, the trace of this matrix, which equals the trace of $T$, is $⟨w,v⟩w$. □