multi-linear

Let $V_1,V_2,\mathrm{\dots },V_n,W$ be vector spaces over a field $K$. Amapping

$$M:V_1\times V_2\times \mathrm{\cdots }\times V_n\to W$$

iscalled multi-linear or $n$-linear, if $M$ is linear in each ofits arguments.

Notes.

• A bilinear mapping is another name for a $2$-linear mapping.

• This definition generalizes in an obvious way to rings andmodules.

• An excellent example of a multi-linear map is the determinant operation.