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In [[multilinear algebra]], a '''multilinear form''' is a [[Map (mathematics)|map]] of the type
:<math>f: V^n \to K , </math>
where ''V'' is a [[vector space]] over the [[field (mathematics)|field]] ''K'', that is separately [[linear]] in each its ''n'' variables.<ref>{{MathWorld |title=Multilinear Form |urlname=MultilinearForm}}</ref>


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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 (algebra)|characteristic]] other than 2, this is equivalent to saying that  
:<math>f(\dots,x,\dots,x,\dots) = 0 , </math>  
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 form]]s.
 
==See also==
*[[Multilinear map]]
*[[Homogeneous polynomial]]
 
==References==
{{Reflist}}
 
{{DEFAULTSORT:Multilinear Form}}
[[Category:Multilinear algebra]]

Latest revision as of 12:42, 13 March 2013

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

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

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

References

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