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

{}\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

.