tensorproduct

The tensor product of two vector spaces V and W over a field F, denoted V ⊗ W (or V ⊗_F W), is a vector space equipped with a bilinear map t: V × W → V ⊗ W that is universal for bilinear maps. The universal property states that for any vector space X and any bilinear map b: V × W → X, there exists a unique linear map φ: V ⊗ W → X such that b(v, w) = φ(v ⊗ w) for all v ∈ V and w ∈ W.

Concretely, if {v_i} is a basis of V and {w_j} is a basis of W, then the elements v_i ⊗ w_j form a basis of V ⊗ W, so dim(V ⊗ W) = dim(V) · dim(W). In particular, if V ≅ F^m and W ≅ F^n, then V ⊗ W ≅ F^{mn}.

The tensor product is associative up to a canonical isomorphism: (V ⊗ W) ⊗ U ≅ V ⊗ (W ⊗ U). There is also a natural commutativity (symmetry) isomorphism V ⊗ W ≅ W ⊗ V. The tensor product is functorial in both arguments and, for rings R, modules V and W over R, one considers V ⊗_R W with the analogous universal property.

It is a central construction in multilinear algebra and has applications across mathematics and physics, including the construction of exterior and symmetric algebras and the description of composite systems in quantum mechanics.