Dual basis

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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:


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).


[a1] P.M. Cohn, "Algebra" , 1 , Wiley (1982)
How to Cite This Entry:
Dual basis. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by E.N. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article