Nilpotent endomorphism/Jordan normal form/Mapping lemma/Fact

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.

Then there exists a

basis ${}v_{1},\ldots ,v_{n}$ of ${}V$ with

${}\varphi (v_{j})=v_{j-1}\,$

or

${}\varphi (v_{j})=0\,.$

Proof, Write another proof