Multilinear form

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

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

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

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

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

• Homogeneous polynomial

References

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