Trigonalizable/Direct sum/Fact

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

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

denote linear mappings, and let

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

denote the product mapping.

Then ${}\varphi$ is

trigonalizable

if and only if this holds for all ${}\varphi _{i}$.

Write a proof