Editing Tensor product (section)

{{Short description|Mathematical operation on vector spaces}}
{{For|generalizations of this concept|Tensor product of modules|Tensor product (disambiguation)}}

In [[mathematics]], the '''tensor product''' <math>V \otimes W</math> of two [[vector space]]s <math>V</math> and <math>W</math> (over the same [[Field (mathematics)|field]]) is a vector space to which is associated a [[bilinear map]] <math>V\times W \rightarrow V\otimes W</math> that maps a pair <math>(v,w),\  v\in V, w\in W</math> to an element of <math>V \otimes W</math> denoted {{tmath|1= v \otimes w }}.

An element of the form <math>v \otimes w</math> is called the '''tensor product''' of <math>v</math> and <math>w</math>. An element of <math>V \otimes W</math> is a [[tensor]], and the tensor product of two vectors is sometimes called an ''elementary tensor'' or a ''decomposable tensor''. The elementary tensors [[linear span|span]] <math>V \otimes W</math> in the sense that every element of <math>V \otimes W</math> is a sum of elementary tensors. If [[basis (linear algebra)|bases]] are given for <math>V</math> and <math>W</math>, a basis of <math>V \otimes W</math> is formed by all tensor products of a basis element of <math>V</math> and a basis element of <math>W</math>.

The tensor product of two vector spaces captures the properties of all bilinear maps in the sense that a bilinear map from <math>V\times W</math> into another vector space <math>Z</math> factors uniquely through a [[linear map]] <math>V\otimes W\to Z</math> (see the section below titled 'Universal property'), i.e. the bilinear map is associated to a unique linear map from the tensor product <math>V \otimes W</math> to <math>Z</math>.

Tensor products are used in many application areas, including physics and engineering. For example, in [[general relativity]], the [[gravitational field]] is described through the [[metric tensor]], which is a [[tensor field]] with one tensor at each point of the [[space-time]] [[manifold]], and each belonging to the tensor product of the [[cotangent space]] at the point with itself.