One approach is to define a tensor as a multilinear map.

A multilinear map of two variables is a bilinear map.

More generally, a multilinear map of k variables is called a k-linear map.

They can be thought of as alternating, multilinear maps on k tangent vectors.

By definition a multilinear map is alternating if it vanishes whenever two of its arguments are equal.

Mathematically, tensors are generalised linear operators - multilinear maps.

A multilinear map has a value of zero whenever one of its arguments is zero.

When described as multilinear maps, the tensor product simply multiplies the two tensors, i.e.

The tensor-to-vector projection in multilinear subspace learning is a multilinear map as well.

The transformation can be equally well applied to any of the vector spaces on which the multilinear map acts to give another distinct invariance.

Multilinear maps and multilinear forms are fundamental objects of study in multilinear algebra.

If the codomain of a multilinear map is the field of scalars, it is called a multilinear form.

For example, any inner product on a vector space is a multilinear map, as is the cross product of vectors in .

The set of all such multilinear maps forms a vector space, called the tensor product space of type at and denoted by .

In linear algebra, a multilinear map is a function of several variables that is linear separately in each variable.

The bundle of differential forms, at each point, consists of all totally antisymmetric multilinear maps on the tangent space at that point.

Moreover, the universal property of the tensor product gives a 1-to-1 correspondence between tensors defined in this way and tensors defined as multilinear maps.

The second Cayley hyperdeterminant from 1845, which is often called "Det", is a discriminant for a singular point on a scalar valued multilinear map.

However, it may be convenient for notation to consider n-ary functions, as for example multilinear maps (which are not linear maps on the product space, if n 1).

As in the case of tensor products of multilinear maps, the number of variables of their exterior product is the sum of the numbers of their variables.

In the general case a hyperdeterminant is defined as a discriminant for a multilinear map f from finite-dimensional vector spaces V to their underlying field K which may be or .

Moreover, such an array can be realised as the components of some multilinear map T. This motivates viewing multilinear maps as the intrinsic objects underlying tensors.

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.

At , these two vector spaces may be used to construct type tensors, which are real-valued multilinear maps acting on the direct sum of copies of the cotangent space with copies of the tangent space.

If one views the permanent as a map that takes n vectors as arguments, then it is a multilinear map and it is symmetric (meaning that any order of the vectors results in the same permanent).