Trigonalizable/Direct sum/Fact/Proof/Exercise

Let ${\displaystyle {}V_{1},\ldots ,V_{n}}$ denote finite-dimensional vector spaces over the field ${\displaystyle {}K}$, let

${\displaystyle \varphi _{i}\colon V_{i}\longrightarrow V_{i}}$

denote linear mappings, and let

${\displaystyle \varphi =\varphi _{1}\times \cdots \times \varphi _{n}\colon V_{1}\times \cdots \times V_{n}\longrightarrow V_{1}\times \cdots \times V_{n}}$

denote the product mapping. Show that ${\displaystyle {}\varphi }$ is trigonalizable if and only if this holds for all ${\displaystyle {}\varphi _{i}}$.