Exercise 10.A.9

Proof. It is easy to check that $\mathit{Pw}=w$ for every $w\in \mathrm{range}<!--\; FUNCTION\; APPLICATION\; -->P$. Let $u_1,\dots ,u_n$ be a basis of $\mathrm{null}<!--\; FUNCTION\; APPLICATION\; -->P$ and let $w_1,\dots ,w_m$ be a basis of $\mathrm{range}<!--\; FUNCTION\; APPLICATION\; -->P$. Exercise 4 in section 5B implies that joining these two basis we get a basis of $V$. Moreover this is a basis consisting of eigenvectors of $P$. Therefore the eigenvalues of $T$ are $0$, with multiplicity $n$, and $1$, with multiplicity $m$. Thus $\mathrm{trace}<!--\; FUNCTION\; APPLICATION\; -->T=m$. □