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 / c / Solution

For $v\in \mathcal{R}(h^{m+k})$, choose $u$ with $v=h^{m+k}u=h^m(h^{k}u)$. Thus $v\in \mathcal{R}(h^m)$, provingThis is the decreasing-image part of stabilization of images of powers of a linear map. The convention $h^0={\mathrm{id}}_V$ includes $m=0$ and $k=0$.