Past exam of the year 1 of the computer science course of the University of Oxford / 2019 / Digital systems, linear algebra, introduction to formal proof / 5 / e / ii

Set $Q=I-P$. Since $Q^2=I-2P+P^2=I-P=Q$, the previous result applies to this projector and gives $\mathcal{N}(Q)=\mathcal{C}(I-Q)$. Substituting $Q=I-P$ proves $\mathcal{N}(I-P)=\mathcal{C}(P)$. These are the kernel-image identities for complementary projectors.