Nilpotent endomorphism/Basis images/1/Exercise

Let ${}v_{1},\ldots ,v_{6}$ be a basis of the ${}K$-vector space ${}V$, and let ${}\varphi \colon V\rightarrow V$ denote the linear mapping, given by

$v_{1},v_{2}\mapsto v_{4},\,v_{3},v_{5}\mapsto v_{6},\,v_{4}\mapsto 0,\,v_{6}\mapsto v_{2}.$

1. Show that ${}\varphi$ is nilpotent.
2. Determine the minimal ${}s$ satisfying ${}\varphi ^{s}=0$.
3. Determine the kernel of ${}\varphi$.
4. Find a basis of ${}V$ such that the describing matrix of ${}\varphi$, with respect to this basis, is in Jordan normal form.

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