Linear mapping/Nilpotent matrix on finite-dimensional vector space/Nilexponent/Exercise

Let ${}K$ be a field, and let ${}V$ denote a finite-dimensional ${}K$-vector space. Let

$\varphi \colon V\longrightarrow V$

be a nilpotent linear mapping. Show that ${}\varphi ^{n}=0$, where ${}n$ is the dimension of ${}V$.

Create a solution