Multilinear map

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In linear algebra, a multilinear map is a function of several variables that is linear separately in each variable. More precisely, a multilinear map is a function

${\displaystyle f\colon V_{1}\times \cdots \times V_{n}\to W{\text{,}}}$

where ${\displaystyle V_{1},\ldots ,V_{n}}$ (${\displaystyle n\in \mathbb {Z} _{\geq 0}}$) and ${\displaystyle W}$ are vector spaces (or modules over a commutative ring), with the following property: for each ${\displaystyle i}$, if all of the variables but ${\displaystyle v_{i}}$ are held constant, then ${\displaystyle f(v_{1},\ldots ,v_{i},\ldots ,v_{n})}$ is a linear function of ${\displaystyle v_{i}}$.^[1] One way to visualize this is to imagine two orthogonal vectors; if one of these vectors is scaled by a factor of 2 while the other remains unchanged, the cross product likewise scales by a factor of two. If both are scaled by a factor of 2, the cross product scales by a factor of ${\displaystyle 2^{2}}$.

A multilinear map of one variable is a linear map, and of two variables is a bilinear map. More generally, for any nonnegative integer ${\displaystyle k}$, a multilinear map of k variables is called a k-linear map. If the codomain of a multilinear map is the field of scalars, it is called a multilinear form. Multilinear maps and multilinear forms are fundamental objects of study in multilinear algebra.

If all variables belong to the same space, one can consider symmetric, antisymmetric and alternating k-linear maps. The latter two coincide if the underlying ring (or field) has a characteristic different from two, else the former two coincide.