Tensor product

In mathematics, the tensor product V ⊗ W {\displaystyle V\otimes W} of two vector spaces V {\displaystyle V} and W {\displaystyle W} (over the same field) is a vector space to which is associated a bilinear map V × W → V ⊗ W {\displaystyle V\times W\rightarrow V\otimes W} that maps a pair ( v , w ) {\displaystyle (v,w)} , where v ∈ V , w ∈ W {\displaystyle v\in V,w\in W} , to an element of V ⊗ W {\displaystyle V\otimes W} denoted ⁠ v ⊗ w {\displaystyle v\otimes w} ⁠. An element of the form v ⊗ w {\displaystyle v\otimes w} is called the tensor product of v {\displaystyle v} and w {\displaystyle w} . An element of V ⊗ W {\displaystyle V\otimes W} is a tensor, and the tensor product of two vectors is sometimes called an elementary tensor or a decomposable tensor. The elementary tensors span V ⊗ W {\displaystyle V\otimes W} in the sense that every element of V ⊗ W {\displaystyle V\otimes W} is a sum of elementary tensors. If bases are given for V {\displaystyle V} and W {\displaystyle W} , a basis of V ⊗ W {\displaystyle V\otimes W} is formed by all tensor products of a basis element of V {\displaystyle V} and a basis element of W {\displaystyle W} . The tensor product of two vector spaces captures the properties of all bilinear maps in the sense that a bilinear map from V × W {\displaystyle V\times W} into another vector space Z {\displaystyle Z} factors uniquely through a linear map V ⊗ W → Z {\displaystyle V\otimes W\to Z} (see § Universal property), i.e. the bilinear map is associated to a unique linear map from the tensor product V ⊗ W {\displaystyle V\otimes W} to Z {\displaystyle Z}...