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Tensor product

The tensor product V ⊗ F W, {\displaystyle V\otimes _{F}W,} or simply V ⊗ W, {\displaystyle V\otimes W,} of two vector spaces V {\displaystyle V} and W {\displaystyle W} is one of the central notions of multilinear algebra which deals with extending notions such as linear maps to several variables. A map g: V × W → X {\displaystyle g:V\times W\to X} from the Cartesian product V × W {\displaystyle V\times W} is called bilinear if g {\displaystyle g} is linear in both variables v {\displaystyle \mathbf {v} } and w. {\displaystyle \mathbf {w}.} That is to say, for fixed w {\displaystyle \mathbf {w} } the map v ↦ g (v, w) {\displaystyle \mathbf {v} \mapsto g(\mathbf {v},\mathbf {w})} is linear in the sense above and likewise for fixed v. {\displaystyle \mathbf {v}.}

The tensor product is a particular vector space that is a universal recipient of bilinear maps g, {\displaystyle g,} as follows. It is defined as the vector space consisting of finite (formal) sums of symbols called tensors v 1 ⊗ w 1 + v 2 ⊗ w 2 + ⋯ + v n ⊗ w n, {\displaystyle \mathbf {v} _{1}\otimes \mathbf {w} _{1}+\mathbf {v} _{2}\otimes \mathbf {w} _{2}+\cdots +\mathbf {v} _{n}\otimes \mathbf {w} _{n},} subject to the rules a ⋅ (v ⊗ w) = (a ⋅ v) ⊗ w = v ⊗ (a ⋅ w), where a is a scalar (v 1 + v 2) ⊗ w = v 1 ⊗ w + v 2 ⊗ w v ⊗ (w 1 + w 2) = v ⊗ w 1 + v ⊗ w 2. {\displaystyle {\begin{alignedat}{6}a\cdot (\mathbf {v} \otimes \mathbf {w})~&=~(a\cdot \mathbf {v})\otimes \mathbf {w} ~=~\mathbf {v} \otimes (a\cdot \mathbf {w}),&&~~{\text{ where }}a{\text{ is a scalar}}\\(\mathbf {v} _{1}+\mathbf {v} _{2})\otimes \mathbf {w} ~&=~\mathbf {v} _{1}\otimes \mathbf {w} +\mathbf {v} _{2}\otimes \mathbf {w} &&\\\mathbf {v} \otimes (\mathbf {w} _{1}+\mathbf {w} _{2})~&=~\mathbf {v} \otimes \mathbf {w} _{1}+\mathbf {v} \otimes \mathbf {w} _{2}.&&\\\end{alignedat}}} These rules ensure that the map f {\displaystyle f} from the V × W {\displaystyle V\times W} to V ⊗ W {\displaystyle V\otimes W} that maps a tuple (v, w) {\displaystyle (\mathbf {v},\mathbf {w})} to v ⊗ w {\displaystyle \mathbf {v} \otimes \mathbf {w} } is bilinear. The universality states that given any vector space X {\displaystyle X} and any bilinear map g: V × W → X, {\displaystyle g:V\times W\to X,} there exists a unique map u, {\displaystyle u,} shown in the diagram with a dotted arrow, whose composition with f {\displaystyle f} equals g: {\displaystyle g:} u (v ⊗ w) = g (v, w). {\displaystyle u(\mathbf {v} \otimes \mathbf {w})=g(\mathbf {v},\mathbf {w}).} This is called the universal property of the tensor product, an instance of the method—much used in advanced abstract algebra—to indirectly define objects by specifying maps from or to this object.