Abstractly, the tensor direct product is the same as the vector space tensor product. However, it reflects an approach toward calculation using coordinates, and indices in particular. The notion of tensor product is more algebraic, intrinsic, and abstract. For instance, up to isomorphism, the tensor product is commutative because V⊗W≅W⊗V. Note this does not mean that the tensor product is symmetric.

direct product | Kronecker product | vector space tensor product