Multilinear map

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 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 where V 1 , … , V n {displaystyle V_{1},ldots ,V_{n}} and W {displaystyle W} are vector spaces (or modules over a commutative ring), with the following property: for each i {displaystyle i} , if all of the variables but v i {displaystyle v_{i}} are held constant, then f ( v 1 , … , v n ) {displaystyle f(v_{1},ldots ,v_{n})} is a linear function of v i {displaystyle v_{i}} . A multilinear map of one variable is a linear map, and of two variables is a bilinear map. More generally, 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 coincide if the underlying ring (or field) has a characteristic different from two, else the former two coincide.