2x2-matrix/Nilpotent/Jordan/Example

A matrix of the form

${}M={\begin{pmatrix}0&a\\0&0\end{pmatrix}}\,$

with ${}a\neq 0$ has, with respect the basis ${}ae_{1}$ and ${}e_{2}$, the form

${\begin{pmatrix}0&1\\0&0\end{pmatrix}}.$