Linear mapping/Tensor product/Multilinearity/Exercise

Let ${}K$ be a field, and let ${}V_{1},\ldots ,V_{n},W_{1},\ldots ,W_{n}$ be vector spaces over ${}K$. Show that the mapping

$\operatorname {Hom} _{K}{\left(V_{1},W_{1}\right)}\times \cdots \times \operatorname {Hom} _{K}{\left(V_{n},W_{n}\right)}\longrightarrow \operatorname {Hom} _{K}{\left(V_{1}\otimes _{}\cdots \otimes _{}V_{n},W_{1}\otimes _{}\cdots \otimes _{}W_{n}\right)},(\varphi _{1},\ldots ,\varphi _{n})\longmapsto \varphi _{1}\otimes _{}\cdots \otimes _{}\varphi _{n},$

is multilinear.

Create a solution