space operation The tensor product of two vector spaces and is commutative in the sense that there is a canonical isomorphism that maps to On the other hand, even when the tensor product of vectors is not commutative; that is in general. The map from to itself induces a linear automorphism that is called a . More generally and as usual (see tensor algebra), let denote the tensor product of copies of the vector space . For every permutation of the first positive integers, the map induces a linear automorphism of which is called a braiding map. Tensor product of linear maps Given a linear map and a vector space , the tensor product is the unique linear map such that The tensor product is defined similarly. Given two linear maps and their tensor product is the unique linear map that satisfies One has In terms of category theory, this means that the tensor product is a bifunctor from the category of vector spaces to itself. If and are both injective or surjective, then the same is true for all above defined linear maps. In particular, the tensor product with a vector space is an exact functor; this means that every exact sequence is mapped to an exact sequence (tensor products of modules do not transform injections into injections, but they are right exact functors). By choosing bases of all vector spaces involved, the linear maps and can be represented by matrices. Then, depending on how the tensor is vectorized, the matrix describing the tensor product is the Kronecker product of the two matrices. For example, if , and above are all two-dimensional and bases have been fixed for all of them, and and are given by the matrices respectively, then the tensor product of these two matrices is The resultant rank is at most 4, and thus the resultant dimension is 4. Note that here denotes the tensor rank i.e. the number of requisite indices (while the matrix rank counts the number of degrees of freedom in the resulting array). Note A dyadic product is the special case of the tensor product between two vectors of the same dimension. General tensors For non-negative integers and a type tensor on a vector space is an element of Here is the dual vector space (which consists of all linear maps from to the ground field ). There is a product map,