"total order" will be used for the total dimension of the array (or its generalization in other definitions), in the preceding example, and the term "type" for the pair giving the number of contravariant and covariant indices. A tensor of type is also called a -tensor for short. This discussion motivates the following formal definition: The definition of a tensor as a multidimensional array satisfying a transformation law traces back to the work of Ricci. An equivalent definition of a tensor uses the representations of the general linear group. There is an action of the general linear group on the set of all ordered bases of an n-dimensional vector space. If is an ordered basis, and is an invertible matrix, then the action is given by Let F be the set of all ordered bases. Then F is a principal homogeneous space for GL(n). Let W be a vector space and let be a representation of GL(n) on W (that is, a group homomorphism ). Then a tensor of type is an equivariant map . Equivariance here means that When is a tensor representation of the general linear group, this gives the usual definition of tensors as multidimensional arrays. This definition is often used to describe tensors on manifolds, and readily generalizes to other groups. As multilinear maps A downside to the definition of a tensor using the multidimensional array approach is that it is not apparent from the definition that the defined object is indeed basis independent, as is expected from an intrinsically geometric object. Although it is possible to show that transformation laws indeed ensure independence from the basis, sometimes a more intrinsic definition is preferred. One approach that is common in differential geometry is to define tensors relative to a fixed (finite-dimensional) vector space V, which is usually taken to be a particular vector space of some geometrical significance like the tangent space to a manifold. In this approach, a type tensor T is defined as a multilinear map, where V∗ is the corresponding dual space of covectors, which is linear in each of its arguments. The above assumes V is a vector space over the real numbers, ℝ. More generally, V can be taken over any field F (e.g. the complex numbers), with F replacing ℝ as the codomain of the multilinear maps. By applying a multilinear map T of type to a basis {ej} for