Nilpotent endomorphism/Basis images/2/Exercise

Let ${}v_{1},\ldots ,v_{7}$ be a basis of the ${}\mathbb {R}$-vector space ${}V$, and let ${}\varphi \colon V\rightarrow V$ denote the linear mapping given by

$v_{3},v_{2}\mapsto v_{5},\,v_{5}\mapsto 4v_{6},\,v_{1},v_{4}\mapsto 0,\,v_{6}\mapsto 5v_{4},\,v_{7}\mapsto 3v_{2}.$

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

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