Homomorphism space/Evaluation at a vector/Linear/Exercise

Let ${}K$ be a field, and let ${}V$ and ${}W$ be ${}K$-vector spaces. Let ${}\operatorname {Hom} _{K}{\left(V,W\right)}$ be the ${}K$-vector space of all linear mappings from ${}V$ to ${}W$, and let ${}v\in V$ denote a fixed vector. Show that the mapping

$F\colon \operatorname {Hom} _{K}{\left(V,W\right)}\longrightarrow W,\varphi \longmapsto F(\varphi ):=\varphi (v),$

is ${}K$-linear.

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