Multilinear mapping/Distributivity law/Fact/Proof/Exercise

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

Φ : V_{1} \times \dots \times V_{n} ⟶ W

be a multilinear mapping, and let $v_{i 1}, \dots, v_{i m_{i}} \in V_{i}$ and $a_{i j} \in K$ for $i = 1, \dots, n$ and $j = 1, \dots, m_{i}$ . Show that

Φ (\sum_{j = 1}^{m_{1}} a_{1 j} v_{1 j}, \dots, \sum_{j = 1}^{m_{n}} a_{n j} v_{n j}) = \sum_{(j_{1}, \dots, j_{n}) \in {1, \dots, m_{1}} \times \dots \times {1, \dots, m_{n}}} a_{1 j_{1}} \dots a_{n j_{n}} Φ (v_{1 j_{1}}, \dots, v_{n j_{n}})

holds.

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