Multilinearform
A multi - linear form is in mathematics a function of vector spaces assigns a value and arguments in each component is linear. In the more general case that the image space is itself a vector space, or image and target spaces are moduli, it is called a multi- linear mapping.
Definition
A picture
Called multi- linear form, if for all and all of the following two conditions are met:
For all
And for all
The set of all multi- linear maps form a vector space. In case you write.
Alternating multi- linear forms
A multi- linear form is called alternating if it is zero, if twice the same vector is used, ie
For everyone.
In this case, it follows that the form is skew-symmetric, that is, that it changes sign when interchanging of any two arguments, ie
For all and. The reverse implication - that all skew-symmetric multilinear forms are alternately - applies only if the characteristic of not 2, so for example.
More generally, an arbitrary permutation of the indices, then applies
Where the sign of the permutation called.
The set of all alternating multilinear forms is a subspace of. In addition, can be applied to this amount, the structure of an algebra define. This algebra is called Grassmann algebra. Important is the special case. Then a 1- dimensional subspace of, and its elements are called determinants features.