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 / b

Take $V={\mathbb{R}}^2$ and the linear map $h(x,y)=(y,0)$. ThenTheir intersection is nonzero and their sum is only one-dimensional. Hence $V\ne \mathcal{R}(h)\oplus \mathcal{N}(h)$. This nilpotent linear map has $h^2=0$, illustrating why taking sufficiently high powers changes the situation.