Linear mapping/Composed with multilinear/Multilinear/Exercise

Let ${}K$ be a field, and let ${}V_{1},\ldots ,V_{n},W_{1},\ldots ,W_{n}$ and ${}Z$ be vector spaces over ${}K$. Let

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

denote linear mappings, and let

$\pi \colon W_{1}\times \cdots \times W_{n}\longrightarrow Z$

be a multilinear mapping. Show that the mapping

$\pi \circ (\varphi _{1}\times \cdots \times \varphi _{n})\colon V_{1}\times \cdots \times V_{n}\longrightarrow Z,(v_{1},\ldots ,v_{n})\longmapsto \pi (\varphi _{1}(v_{1}),\ldots ,\varphi _{n}(v_{n})),$

is also multilinear.

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