Multilinearity

Redirect to:

• Multilinear algebra

Multilinearity is linearity spread, independently, among several variables.

Suppose $X$ is a vector space and suppose we have a function: $$\varphi \colon X^k \rightarrow {\bf R},$$ $$\varphi (x,y,z,...)= r \in {\bf R}.$$

It is called multilinear if

• $\varphi(\cdot, b, c, ...)$ is linear for each $b, c,...$,
• $\varphi(a, \cdot, c, ...)$ is linear for each $a, c,...$,
• $\varphi(a, b, \cdot, ...)$ is linear for each $a, b,...$,
• etc.

As an example, see differential forms.