Past exam of the year 1 of the computer science course of the University of Oxford / 2023 / Digital systems, linear algebra and introduction to formal proof / 5 / d / Solution

Apply $h$ to both sides of $\mathcal{R}(h^m)=\mathcal{R}(h^{m+1})$. Because $h(\mathcal{R}(h^j))=\mathcal{R}(h^{j+1})$, we obtain $\mathcal{R}(h^{m+1})=\mathcal{R}(h^{m+2})$. Repeating this argument proves equality at every later step. More formally, an induction on $k$ givesThe base case is $k=0$; the induction step follows from applying the linear map $h$ and then the assumed first equality. Thus one equality permanently stabilizes the descending sequence of images of a linear map.