Multilinear form

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

$f:V^{n}\to K$

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:

$f(\dots ,x,\dots ,x,\dots )=0.$

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:

$f(\dots ,x,\dots ,y,\dots )=-f(\dots ,y,\dots ,x,\dots ).$

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.

See also

• Bilinear map
• Exterior algebra
• Homogeneous polynomial
• Linear map
• Multilinear algebra
• Multilinear map

References

1. Weisstein, Eric W. "Multilinear Form". MathWorld.