Trigonalizable/Direct sum/Fact/Proof/Exercise

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. Show that ${}\varphi$ is trigonalizable if and only if this holds for all ${}\varphi _{i}$.

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