Tensor product

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the tensor product 
 
 
 
 V
 ⊗
 W
 
 
 {\displaystyle V\otimes W}
 
 of two <a href="/facts/Vector_space/M1pxMLx2">vector spaces</a> 
 
 
 
 V
 
 
 {\displaystyle V}
 
 and 
 
 
 
 W
 
 
 {\displaystyle W}
 
 (over the same <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a>) is a vector space to which is associated a <a href="/facts/Bilinear_map/LH1lHVeP">bilinear map</a> 
 
 
 
 V
 ×
 W
 →
 V
 ⊗
 W
 
 
 {\displaystyle V\times W\rightarrow V\otimes W}
 
 that maps a pair 
 
 
 
 (
 v
 ,
 w
 )
 ,
  
 v
 ∈
 V
 ,
 w
 ∈
 W
 
 
 {\displaystyle (v,w),\ 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 <a href="/facts/Tensor/pNjFo5tV">tensor</a>, and the tensor product of two vectors is sometimes called an elementary tensor or a decomposable tensor. The elementary tensors <a href="/facts/Linear_span/vPjclEtb">span</a> 
 
 
 
 V
 ⊗
 W
 
 
 {\displaystyle V\otimes W}
 
 in the sense that every element of 
 
 
 
 V
 ⊗
 W
 
 
 {\displaystyle V\otimes W}
 
 is a sum of elementary tensors. If <a href="/facts/Basis_(linear_algebra)/89IPoN6c">bases</a> are given for 
 
 
 
 V
 
 
 {\displaystyle V}
 
 and 
 
 
 
 W
 
 
 {\displaystyle W}
 
, a basis of 
 
 
 
 V
 ⊗
 W
 
 
 {\displaystyle V\otimes W}
 
 is formed by all tensor products of a basis element of 
 
 
 
 V
 
 
 {\displaystyle V}
 
 and a basis element of 
 
 
 
 W
 
 
 {\displaystyle W}
 
.
The tensor product of two vector spaces captures the properties of all bilinear maps in the sense that a bilinear map from 
 
 
 
 V
 ×
 W
 
 
 {\displaystyle V\times W}
 
 into another vector space 
 
 
 
 Z
 
 
 {\displaystyle Z}
 
 factors uniquely through a <a href="/facts/Linear_map/Q1PBKNVV">linear map</a> 
 
 
 
 V
 ⊗
 W
 →
 Z
 
 
 {\displaystyle V\otimes W\to Z}
 
 (see the section below titled 'Universal property'), i.e. the bilinear map is associated to a unique linear map from the tensor product 
 
 
 
 V
 ⊗
 W
 
 
 {\displaystyle V\otimes W}
 
 to 
 
 
 
 Z
 
 
 {\displaystyle Z}
 
.
Tensor products are used in many application areas, including physics and engineering. For example, in <a href="/facts/General_relativity/jVsoIg8X">general relativity</a>, the <a href="/facts/Gravitational_field/K1uTmJFy">gravitational field</a> is described through the <a href="/facts/Metric_tensor/tbqek1gJ">metric tensor</a>, which is a <a href="/facts/Tensor_field/cAC6F6rF">tensor field</a> with one tensor at each point of the <a href="/facts/Space-time/uRIN1x4b">space-time</a> <a href="/facts/Manifold/rXPERJtu">manifold</a>, and each belonging to the tensor product of the <a href="/facts/Cotangent_space/INNIyG4S">cotangent space</a> at the point with itself.

Tensor product open-in-new

Tensor product