# Multilinear form

Jump to navigation
Jump to search

In mathematics, more specifically in abstract algebra and multilinear algebra, a **multilinear form** is a map of the type

where *V* is a vector space over the field *K*, which is separately linear in each of its *n* variables.^{[1]}

For *n* = 2, i.e. only two variables, one calls ƒ a bilinear form.

An important type of multilinear forms are **alternating multilinear forms**, which have the additional property of changing their sign under exchange of two arguments. When *K* has characteristic other than 2, this is equivalent to saying that

i.e. the form vanishes if supplied the same argument twice. (The exceptional case of characteristic 2 requires more care.) Special cases of these are determinant forms and differential forms.