In abstract algebra and multilinear algebra, a multilinear form on a vector space over a field is a map that is separately -linear in each of its arguments. More generally, one can define multilinear forms on a module over a commutative ring. The rest of this article, however, will only consider multilinear forms on finite-dimensional vector spaces. A multilinear -form on over is called a (covariant) -tensor, and the vector space of such forms is usually denoted or . Tensor product Given a -tensor and an -tensor , a product , known as the tensor product, can be defined by the property for all . The tensor product of multilinear forms is not commutative; however it is bilinear and associative: , and If forms a basis for an -dimensional vector space and is the corresponding dual basis for the dual space , then the products , with form a basis for . Consequently, has dimensionality . Examples Bilinear forms If , is referred to as a bilinear form. A familiar and important example of a (symmetric) bilinear form is the standard inner product (dot product) of vectors. Alternating multilinear forms An important class of multilinear forms are the alternating multilinear forms, which have the additional property that where is a permutation and denotes its sign (+1 if even, –1 if odd). As a consequence, alternating multilinear forms are antisymmetric with respect to swapping of any two arguments (i.e., and ): With the additional hypothesis that the characteristic of the field is not 2, setting implies as a corollary that ; that is, the form has a value of 0 whenever two of its arguments are equal. Note, however, that some authors use this last condition as the defining property of alternating forms. This definition implies the property given at the beginning of the section, but as noted above, the converse implication holds only when . An alternating multilinear -form on over is called a multicovector of degree or -covector, and the vector space of such alternating forms, a subspace of , is generally denoted , or, using the notation for the isomorphic kth exterior power of (the dual space of ), . Note that linear functionals (multilinear 1-forms over ) are trivially alternating, so that , while, by convention, 0-forms are defined to be scalars: . The determinant on matrices, viewed as an argument function of the column vectors, is