Linear mappings/Product mapping/Linear/Exercise

Let ${}K$ be a field. For ${}i\in \{1,\ldots ,n\}$, let ${}K$-vector spaces ${}V_{i}$ and ${}W_{i}$, and linear mappings

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

be given. Show that the product mapping

$\varphi =\varphi _{1}\times \varphi _{2}\times \cdots \times \varphi _{n}\colon V_{1}\times V_{2}\times \cdots \times V_{n}\longrightarrow W_{1}\times W_{2}\times \cdots \times W_{n},\left(v_{1},\,v_{2},\,\ldots ,\,v_{n}\right)\longmapsto \left(\varphi _{1}(v_{1}),\,\varphi _{2}(v_{2}),\,\ldots ,\,\varphi _{n}(v_{n})\right),$

is also a linear mapping between the product spaces.

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