Past exam of the mathematics course of the University of Cambridge / 2020 / ib / Paper 2 / 18H / d

Because $\mathrm{col}Z\subseteq \mathrm{col}X$, the two orthogonal projection matrices satisfyConsequently $P-P_0$ is itself an orthogonal projection, onto the orthogonal complement of $\mathrm{col}Z$ within $\mathrm{col}X$. In particular it is a positive semidefinite matrix, so for every $u\in {\mathbb{R}}^n$,