Multilinear mapping/Generating system/Exercise

Let ${}K$ be a field, and let ${}V_{1},\ldots ,V_{n}$ and ${}W$ denote vector spaces over ${}K$. Let ${}v_{i_{j}}$, ${}i_{j}\in I_{j}$, be generating systems of ${}V_{j}$, ${}j=1,\ldots ,n$. Show that a multilinear mapping

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

is determined by

$\triangle (v_{i_{1}},\ldots ,v_{i_{n}}).$

Create a solution