K^n/Inner product/Bilinear/Exercise

Let ${}K$ be a field, and let ${}n\in \mathbb {N}$. Show that the mapping

$K^{n}\times K^{n}\longrightarrow K,({\begin{pmatrix}u_{1}\\\vdots \\u_{n}\end{pmatrix}},{\begin{pmatrix}v_{1}\\\vdots \\v_{n}\end{pmatrix}})\longmapsto (u_{1},\ldots ,u_{n})\circ {\begin{pmatrix}v_{1}\\\vdots \\v_{n}\end{pmatrix}},$

is multilinear.

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