Multilinear mapping/Distributivity law/Fact/Proof/Exercise

Let ${}K$ be a field, and let ${}V_{1},\ldots ,V_{n}$ and ${}W$ be vector spaces over ${}K$. Let

$\Phi \colon V_{1}\times \cdots \times V_{n}\longrightarrow W$

be a multilinear mapping, and let ${}v_{1j},\ldots ,v_{m_{j}j}\in V_{j}$ and ${}a_{ij}\in K$. Show that

$\Phi {\left(\sum _{i=1}^{m_{1}}a_{i1}v_{i1},\ldots ,\sum _{i=1}^{m_{n}}a_{in}v_{in}\right)}=\sum _{(i_{1},\ldots ,i_{n})\in \{1,\ldots ,m_{1}\}\times \cdots \times \{1,\ldots ,m_{n}\}}a_{i_{1}}\cdots a_{i_{n}}\Phi {\left(v_{i_{1}1},\ldots ,v_{i_{n}n}\right)}\,$

holds.

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