# Dual basis

*to a basis of a module with respect to a form *

A basis of such that

where is a free -module over a commutative ring with a unit element, and is a non-degenerate (non-singular) bilinear form on .

Let be the dual module of , and let be the basis of dual to the initial basis of : , , . To each bilinear form on there correspond mappings , defined by the equations

If the form is non-singular, are isomorphisms, and vice versa. Here the basis dual to is distinguished by the following property:

#### Comments

A bilinear form on is non-degenerate (also called non-singular) if for all , for all implies and for all , for all implies . Occasionally the terminology conjugate module (conjugate space) is used instead of dual module (dual space).

#### References

[a1] | P.M. Cohn, "Algebra" , 1 , Wiley (1982) |

**How to Cite This Entry:**

Dual basis. E.N. Kuz'min (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Dual_basis&oldid=14785