tensor product basis

The following theoremMathworldPlanetmath describes a basis of the tensor product (http://planetmath.org/TensorProduct) of two vector spacesMathworldPlanetmath, 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.


tensor product

  • Theorem.ย ย Let U and V be vector spaces over a field ๐’ฆ with bases


    respectively. Then

    {๐žiโŠ—๐Ÿj}(i,j)โˆˆIร—J (1)

    is a basis for the tensor product space UโŠ—V.



W={ฯˆ:Iร—JโŸถ๐’ฆโขย .f-1โข(๐’ฆโˆ–{0})โขย is finite}โข;

this set is obviously a ๐’ฆ-vector-space under pointwise addition and multiplicationPlanetmathPlanetmath by scalar (see also this (http://planetmath.org/FreeVectorSpaceOverASet) article). Let p:Uร—VโŸถW be the bilinear map which satisfies

pโข(๐ži,๐Ÿj)โข(k,l)={1ifย i=kย andย j=l,0otherwise (2)

for all i,kโˆˆI and j,lโˆˆJ, i.e., pโข(๐ži,๐Ÿj)โˆˆW is the characteristic functionMathworldPlanetmathPlanetmathPlanetmath of {(i,j)}. The reasons (2) uniquely defines p on the whole of Uร—V are that {๐ži}iโˆˆI is a basis of U, {๐Ÿi}jโˆˆJ is a basis of V, and p is bilinear.

Observe that


is a basis of W. Since one may always define a linear map by giving its values on the basis elements, this implies that there for every ๐’ฆ-vector-space X and every map ฮณ:Uร—VโŸถX exists a unique linear map ฮณ^:WโŸถX such that

ฮณ^โข(pโข(๐ži,๐Ÿj))=ฮณโข(๐ži,๐Ÿj)โ€ƒfor allย iโˆˆIย andย jโˆˆJ.

For ฮณ that are bilinear it holds for arbitrary ๐ฎ=โˆ‘iโˆˆIโ€ฒuiโข๐žiโˆˆU and ๐ฏ=โˆ‘jโˆˆJโ€ฒvjโข๐ŸjโˆˆV that ฮณโข(๐ฎ,๐ฏ)=(ฮณ^โˆ˜p)โข(๐ฎ,๐ฏ), since


As this is the defining property of the tensor product UโŠ—V however, it follows that W is (an incarnation of) this tensor product, with ๐ฎโŠ—๐ฏ:=pโข(๐ฎ,๐ฏ). Hence the claim in the theorem is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to the observation about the basis of W. โˆŽ

Title tensor product basis
Canonical name TensorProductBasis
Date of creation 2013-03-22 15:24:48
Last modified on 2013-03-22 15:24:48
Owner lars_h (9802)
Last modified by lars_h (9802)
Numerical id 11
Author lars_h (9802)
Entry type Theorem
Classification msc 15A69
Synonym basis construction of tensor product
Related topic TensorProduct
Related topic FreeVectorSpaceOverASet