Multilinear form: Difference between revisions
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antisymmetric forms only change sign when you exchange adjacent pairs of arguments. The article previously said this holds when you exchange any pair of arguments. |
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Special cases of these are [[determinant]] forms and [[differential form]]s. |
Special cases of these are [[determinant]] forms and [[differential form]]s. |
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An alternating multilinear form is also antisymmetric, where the form changes sign under exchange of any pair of arguments: |
An alternating multilinear form is also antisymmetric, where the form changes sign under exchange of any adjacent pair of arguments: |
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:<math>f(\dots,x |
:<math>f(\dots,x,y,\dots) = -f(\dots,y,x,\dots) .</math> |
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This holds even when the [[Characteristic (algebra)|characteristic]] is 2, though in this case antisymmetry is equivalent to symmetry. Conversely, an antisymmetric form is not necessarily alternating in characteristic 2. |
This holds even when the [[Characteristic (algebra)|characteristic]] is 2, though in this case antisymmetry is equivalent to symmetry. Conversely, an antisymmetric form is not necessarily alternating in characteristic 2. |
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Revision as of 09:17, 17 November 2016
In mathematics, more specifically in abstract algebra and multilinear algebra, a multilinear form is a map of the type
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:
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 adjacent pair of arguments:
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.