In abstract algebra and multilinear algebra, a multilinear form is a map of the type
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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.
Bilinear form
For n = 2, i.e. only two variables, f is referred to as a bilinear form.
Differential form
A differential form is a multilinear form with differential operators.
Alternating multilinear 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 also has the property of antisymmetry, where the form changes sign under exchange of any pair of arguments:
This antisymmetry holds even when the characteristic is 2, although with characteristic 2 antisymmetry is equivalent to symmetry. Conversely, an antisymmetric form is not necessarily alternating in characteristic 2.