Tensor product
Mathematical operation on vector spaces / From Wikipedia, the free encyclopedia
In mathematics, the tensor product of two vector spaces V and W (over the same field) is a vector space to which is associated a bilinear map
that maps a pair
to an element of
denoted
.
An element of the form is called the tensor product of v and w. An element of
is a tensor, and the tensor product of two vectors is sometimes called an elementary tensor or a decomposable tensor. The elementary tensors span
in the sense that every element of
is a sum of elementary tensors. If bases are given for V and W, a basis of
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 into another vector space Z factors uniquely through a linear map
(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 (so like an vector field but with tensors instead of vectors), 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.