Tensor product

In mathematics, the tensor product V ⊗ W of two vector spaces V and W (over the same field) is itself a vector space, endowed with the operation of bilinear composition, denoted by ⊗, from ordered pairs in the Cartesian product V × W onto V ⊗ W in a way that generalizes the outer product.The vector space V ⊗ W {displaystyle Votimes W} and the associated bilinear map φ : V × W → V ⊗ W {displaystyle varphi :V imes W o Votimes W} have the property that any bilinear map h : V × W → Z {displaystyle h:V imes W o Z} from V × W {displaystyle V imes W} to any vector space Z {displaystyle Z} factors through φ {displaystyle varphi } uniquely. By saying ' h {displaystyle h} factors through φ {displaystyle varphi } uniquely,' we mean that there is a unique linear map h ~ : V ⊗ W → Z {displaystyle { ilde {h}}:Votimes W o Z} such that h = h ~ ∘ φ {displaystyle h={ ilde {h}}circ varphi } . In mathematics, the tensor product V ⊗ W of two vector spaces V and W (over the same field) is itself a vector space, endowed with the operation of bilinear composition, denoted by ⊗, from ordered pairs in the Cartesian product V × W onto V ⊗ W in a way that generalizes the outer product.