Nilpotent endomorphism/Successive kernels/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. Let

${}\varphi ^{s}=0\,$

and suppose that ${}s$ is minimal with this property.

Then, between the linear subspaces

${}V_{i}:=\operatorname {kern} \varphi ^{i}\,$

the relation

${}\varphi ^{-1}(V_{i})=V_{i+1}\,$

holds, and the inclusions

${}V_{i}\subset V_{i+1}\,$

are strict for

${}i<s$.

Proof, Write another proof