Multilinear form

In multilinear algebra, a multilinear form is a map of the type

<math>f: V^N \to K<math>,

where V is a vector space over the field K, that is separately linear in each its N variables.

As the word "form" usually denotes a mapping from a vector space into its underlying field, the more general term "multilinear map" is used, when talking about a general map that is linear in all its arguments.

For N = 2, i.e. only two variables, one calls f a bilinear form.

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

<math>f(\dots,x,\dots,x,\dots)=0<math>,

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.