Editing Tensor product (section)

== Properties ==
=== Dimension ===
If {{math|''V''}} and {{math|''W''}} are vector spaces of finite [[dimension (linear algebra)|dimension]], then <math>V\otimes W</math> is finite-dimensional, and its dimension is the product of the dimensions of {{math|''V''}} and {{math|''W''}}.

This results from the fact that a basis of <math>V\otimes W</math> is formed by taking all tensor products of a basis element of {{math|''V''}} and a basis element of {{math|''W''}}.

=== Associativity ===

The tensor product is [[associative]] in the sense that, given three vector spaces {{tmath|1= U, V, W }}, there is a canonical isomorphism: 
: <math>(U\otimes V)\otimes W\cong U\otimes (V\otimes W),</math>
that maps <math>(u\otimes v)\otimes w</math> to {{tmath|1= u\otimes (v \otimes w) }}.

This allows omitting parentheses in the tensor product of more than two vector spaces or vectors.

=== Commutativity as vector space operation ===

The tensor product of two vector spaces <math>V</math> and <math>W</math> is [[commutative ]] in the sense that there is a canonical isomorphism: 
: <math> V \otimes W \cong W\otimes V,</math>
that maps <math>v \otimes w</math> to {{tmath|1= w \otimes v }}.

On the other hand, even when {{tmath|1= V=W }}, the tensor product of vectors is not commutative; that is {{tmath|1= v\otimes w \neq w \otimes v }}, in general.

{{anchor|Tensor powers and braiding}}
The map <math>x\otimes y \mapsto y\otimes x</math> from <math>V\otimes V</math> to itself induces a linear [[automorphism]] that is called a '''{{vanchor|braiding map}}'''. 
More generally and as usual (see [[tensor algebra]]), let <math>V^{\otimes n}</math> denote the tensor product of {{mvar|n}} copies of the vector space {{mvar|V}}. For every [[permutation]] {{mvar|s}} of the first {{mvar|n}} positive integers, the map:
: <math>x_1\otimes \cdots \otimes x_n \mapsto x_{s(1)}\otimes \cdots \otimes x_{s(n)}</math>
induces a linear automorphism of {{tmath|1= V^{\otimes n}\to V^{\otimes n} }}, which is called a braiding map.