Multilinear form

From formulasearchengine
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.

See also