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 / i / Solution

Prove both inclusions. If $v$ belongs to the column space of $I-P$, then $v=(I-P)u$ for some $u$ and $Pv=(P-P^2)u=0$. Thus $v\in \mathcal{N}(P)$.

Conversely, if $Pv=0$, then $v=(I-P)v$, which explicitly writes $v$ as an image of a linear map under $I-P$. Hence $v\in \mathcal{C}(I-P)$. The two inclusions prove $\mathcal{N}(P)=\mathcal{C}(I-P)$. No orthogonality assumption was used.