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, which is separately linear in each of its n variables.

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

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