tensor product basis
The following theorem describes a basis of the tensor product (http://planetmath.org/TensorProduct) of two vector spaces, in terms of given bases of the spaces. In passing, it also gives a construction of this tensor product. The exact same method can be used also for free modules over a commutative ring with unit.
Theorem. Let and be vector spaces over a field with bases
is a basis for the tensor product space .
this set is obviously a -vector-space under pointwise addition and multiplication by scalar (see also this (http://planetmath.org/FreeVectorSpaceOverASet) article). Let be the bilinear map which satisfies
is a basis of . Since one may always define a linear map by giving its values on the basis elements, this implies that there for every -vector-space and every map exists a unique linear map such that
For that are bilinear it holds for arbitrary and that , since
As this is the defining property of the tensor product however, it follows that is (an incarnation of) this tensor product, with . Hence the claim in the theorem is equivalent to the observation about the basis of . ∎
|Title||tensor product basis|
|Date of creation||2013-03-22 15:24:48|
|Last modified on||2013-03-22 15:24:48|
|Last modified by||lars_h (9802)|
|Synonym||basis construction of tensor product|