Multilinear form - Citizendia

In 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, that is separately linear in each its N variables. In Mathematics, multilinear algebra extends the methods of Linear algebra. In Mathematics and related technical fields the term map or mapping is often a Synonym for function. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division The word linear comes from the Latin word linearis, which means created by lines.

As the word "form" usually denotes a mapping from a vector space into its underlying field, the more general term "multilinear map" is used, when talking about a general map that is linear in all its arguments. In Linear algebra, a multilinear map is a Mathematical function of several vector variables that is linear in each variable

For N = 2, i. e. only two variables, one calls f a bilinear form. In Mathematics, a bilinear form on a Vector space V is a Bilinear mapping V × V → F, where

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. In Mathematics, the characteristic of a ring R, often denoted char( R) is defined to be the smallest number of times one must add the ring's 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. In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n In the mathematical fields of Differential geometry and Tensor calculus, differential forms are an approach to Multivariable calculus which is

See also