Linear mappings/To product/Exercise

Let ${}K$ be a field, and let ${}U,V,W$ denote vector spaces over ${}K$. Let ${}\varphi \colon U\rightarrow V$ and ${}\psi \colon U\rightarrow W$ be linear mappings. Show that the mapping

$U\longrightarrow V\times W,u\longmapsto \varphi (u)\times \psi (u)$

into the product space ${}V\times W$ is also a linear mapping.

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