Nilpotent mapping/Kernel one-dimensional/Surjectivity/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. Suppose that the kernel of ${}\varphi$ is one-dimensional. Let

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

and let ${}s$ be the minimal number with

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

1. Show that all ${}V_{i}$, ${}1\leq i\leq s$, have a direct decomposition

   ${}V_{i}=V_{i-1}\oplus U_{i}\,,$

   where ${}U_{i}$ is one-dimensional.

2. Show that the restrictions

   $\varphi \colon U_{i}\longrightarrow V_{i-1}$

   are bijective for ${}1<i<s$.

3. Show that ${}s$ equals the dimension of ${}V$.

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