Multilinear form
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 (and more generally, a module over a commutative ring), that is separately K-linear in each of its n arguments.[1]
For n = 2, i.e. only two variables, f is referred to as a bilinear form.
An important type of multilinear forms are alternating multilinear forms, which have the additional property of vanishing if supplied the same argument twice:
Special cases of these are determinant forms and differential forms.
An alternating multilinear form is also antisymmetric, where the form changes sign under exchange of any pair of arguments:
This holds even when the characteristic is 2, though in this case antisymmetry is equivalent to symmetry. Conversely, an antisymmetric form is not necessarily alternating in characteristic 2.