Dual basis
to a basis of a module
with respect to a form
A basis of
such that
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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
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If the form is non-singular,
are isomorphisms, and vice versa. Here the basis
dual to
is distinguished by the following property:
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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) |
Dual basis. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dual_basis&oldid=14785