Let $V$ be a vectorspace, $\dim V=k$ and $\phi:V\to V$ an endomorphism. Further, there is a characteristic polynomial $x\phi(x)=\sum_{i=0}a_ix^i$ of $\phi$ irreducible,i.e. $x\phi=fg\implies \deg f=0\;\lor \deg g=0$
Show that there is a minor basis $B$ of $V$:
\begin{equation} D_{B B}(\Phi)=\left(\begin{array}{ccccccc} {0} & {0} & {\cdots} & {\cdots} & {\cdots} & {0} & {-a_{0}} \\ {1} & {0} & {0} & {\cdots} & {\cdots} & {0} & {-a_{1}} \\ {0} & {1} & {0} & {0} & {\cdots} & {0} & {-a_{2}} \\ {\vdots} & {\ddots} & {\ddots} & {\ddots} & {\ddots} & {\vdots} & {\vdots} \\ {0} & {\cdots} & {0} & {1} & {0} & {0} & {-a_{n-3}} \\ {0} & {\cdots} & {\cdots} & {0} & {1} & {0} & {-a_{n-2}} \\ {0} & {\cdots} & {\cdots} & {\cdots} & {0} & {1} & {-a_{n-1}} \end{array}\right) \in \mathbb{K}^{n \times n} \end{equation}