Tensor product

In mathematics, the tensor product ${\displaystyle V\otimes W}$ of two vector spaces V and W (over the same field) is a vector space to which is associated a bilinear map ${\displaystyle V\times W\rightarrow V\otimes W}$ that maps a pair ${\displaystyle (v,w),\ v\in V,w\in W}$ to an element of ${\displaystyle V\otimes W}$ denoted ⁠${\displaystyle v\otimes w}$⁠.

An element of the form ${\displaystyle v\otimes w}$ is called the tensor product of v and w. An element of ${\displaystyle V\otimes W}$ is a tensor, and the tensor product of two vectors is sometimes called an elementary tensor or a decomposable tensor. The elementary tensors span ${\displaystyle V\otimes W}$ in the sense that every element of ${\displaystyle V\otimes W}$ is a sum of elementary tensors. If bases are given for V and W, a basis of ${\displaystyle V\otimes W}$ is formed by all tensor products of a basis element of V and a basis element of W.

The tensor product of two vector spaces captures the properties of all bilinear maps in the sense that a bilinear map from ${\displaystyle V\times W}$ into another vector space Z factors uniquely through a linear map ${\displaystyle V\otimes W\to Z}$ (see Universal property).

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.