Endomorphism/Trigonalizable/Lower triangular matrix/Exercise

Let ${}K$ denote a field, and let ${}V$ denote a ${}K$-vector space of finite dimension. Let

$\varphi \colon V\longrightarrow V$

be a linear mapping. Show that ${}\varphi$ is trigonalizable if and only if ${}\varphi$ is described by a lower triangular matrix

${\begin{pmatrix}a_{1}&0&\cdots &\cdots &0\\\ast &a_{2}&0&\cdots &0\\\vdots &\ddots &\ddots &\ddots &\vdots \\\ast &\cdots &\ast &a_{n-1}&0\\\ast &\cdots &\cdots &\ast &a_{n}\end{pmatrix}}$

with respect to a suitable basis.

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