Vector spaces/Dual mapping/Total assignement/Linearity/Exercise

Let ${}K$ be a field, and let ${}V$ and ${}W$ denote ${}K$-vector spaces. Show that the mapping

$\operatorname {Hom} _{K}{\left(V,W\right)}\longrightarrow \operatorname {Hom} _{K}{\left({W}^{*},{V}^{*}\right)},\varphi \longmapsto \varphi ^{*},$

which assigns to a linear mapping its dual mapping, is linear.

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