tensor product - definitie. Wat is tensor product
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Wat (wie) is tensor product - definitie

CONCEPT IN LINEAR ALGEBRA, GENERALIZED THROUGHOUT MATHEMATICS
Tensor products; ⊗; Tensor multiplication; Tensor product of vector spaces; Tensor product (vector spaces); Tensor Product; Tensor product representation; The tensor product; Tensor product of linear maps
  • commutative]] (that is, <math>h = \tilde{h} \circ \varphi</math>).

tensor product         
<mathematics> A function of two vector spaces, U and V, which returns the space of linear maps from V's dual to U. Tensor product has natural symmetry in interchange of U and V and it produces an associative "multiplication" on vector spaces. Wrinting * for tensor product, we can map UxV to U*V via: (u,v) maps to that linear map which takes any w in V's dual to u times w's action on v. We call this linear map u*v. One can then show that u * v + u * x = u * (v+x) u * v + t * v = (u+t) * v and hu * v = h(u * v) = u * hv ie, the mapping respects linearity: whence any {bilinear map} from UxV (to wherever) may be factorised via this mapping. This gives us the degree of natural symmetry in swapping U and V. By rolling it up to multilinear maps from products of several vector spaces, we can get to the natural associative "multiplication" on vector spaces. When all the vector spaces are the same, permutation of the factors doesn't change the space and so constitutes an automorphism. These permutation-induced iso-auto-morphisms form a group which is a model of the group of permutations. (1996-09-27)
Tensor product         
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes W denoted v \otimes w.
Tensor product of modules         
  • right
OPERATION THAT PAIRS A LEFT AND A RIGHT 𝑅‐MODULE INTO AN ABELIAN GROUP
Tensor product of modules over a ring; Exterior bundle; Relative tensor product; Tensor product of abelian groups; Balanced product; Trace map; Tensor product of complexes
In mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps (e.g.

Wikipedia

Tensor product

In mathematics, the tensor product V W {\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 V × W V W {\displaystyle V\times W\to 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 and w. An element of V W {\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 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 bases are given for V and W, a basis of V W {\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 V × W {\displaystyle V\times W} into another vector space Z factors uniquely through a linear map V W Z {\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 vector field of tensors, one at each point of the space-time manifold, and each belonging to the tensor product with itself of the cotangent space at the point.