Tensor product

Mathematical operation on vector spaces

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.

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