Multilinear map - Knowlpedia

Let $V_1,\ldots,V_k,W$ be vector spaces over the same field $K$ , with $k\ge1$ . A map $f:V_1\times\cdots\times V_k\to W$ is multilinear if, for every argument position $i$ , fixed values of the other arguments, $u,v\in V_i$ , and $a,b\in K$ ,

f(v_1,\ldots,au+bv,\ldots,v_k) =a f(v_1,\ldots,u,\ldots,v_k)+b f(v_1,\ldots,v,\ldots,v_k).

Terminology and examples

Such a map is also called $k$ -linear. For $k=1$ this is a linear map; for $k=2$ it is a bilinear map.

The product $(x,y,z)\mapsto xyz$ from $K^3$ to $K$ is trilinear. A multilinear map need not be linear as a map from the product vector space: $(x,y)\mapsto xy$ is bilinear but does not preserve addition of pairs.

References

Bernhard Leeb, Some multilinear algebra, 2020, §§1.1, 1.4 and 2.3.