multi-linear

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

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

is called multi-linear or $n$-linear, if $M$ is linear in each of its arguments.

Notes.

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

• This definition generalizes in an obvious way to rings and modules.

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

Title	multi-linear
Canonical name	Multilinear
Date of creation	2013-03-22 12:33:05
Last modified on	2013-03-22 12:33:05
Owner	rmilson (146)
Last modified by	rmilson (146)
Numerical id	10
Author	rmilson (146)
Entry type	Definition
Classification	msc 15A69
Synonym	multi-linearity
Synonym	multilinear
Synonym	multilinearity
Related topic	BilinearForm
Related topic	BilinearMap