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''to a basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034080/d0340801.png" /> of a module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034080/d0340802.png" /> with respect to a form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034080/d0340803.png" />''
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A basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034080/d0340804.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034080/d0340805.png" /> such that
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{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034080/d0340806.png" /></td> </tr></table>
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''to a basis  $  \{ e _ {1}, \dots, e _ {n} \} $ of a module  $  E $ with respect to a form  $  f $''
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034080/d0340807.png" /></td> </tr></table>
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A basis  $  \{ c _ {1}, \dots, c _ {n} \} $
 +
of  $  E $
 +
such that
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034080/d0340808.png" /> is a free <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034080/d0340809.png" />-module over a commutative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034080/d03408010.png" /> with a unit element, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034080/d03408011.png" /> is a non-degenerate (non-singular) bilinear form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034080/d03408012.png" />.
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$$
 +
f ( e _ {i} , c _ {i} )  = 1 ,\  f ( e _ {i} , c _ {j} ) = 0 ,
 +
$$
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034080/d03408013.png" /> be the dual module of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034080/d03408014.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034080/d03408015.png" /> be the basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034080/d03408016.png" /> dual to the initial basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034080/d03408017.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034080/d03408018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034080/d03408019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034080/d03408020.png" />. To each bilinear form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034080/d03408021.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034080/d03408022.png" /> there correspond mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034080/d03408023.png" />, defined by the equations
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$$
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i  \neq  j ,\  1  \leq  i , j  \leq  n ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034080/d03408024.png" /></td> </tr></table>
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where  $  E $
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is a free  $  K $-module over a commutative ring  $  K $
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with a unit element, and  $  f $
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is a non-degenerate (non-singular) bilinear form on  $  E $.
  
If the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034080/d03408025.png" /> is non-singular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034080/d03408026.png" /> are isomorphisms, and vice versa. Here the basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034080/d03408027.png" /> dual to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034080/d03408028.png" /> is distinguished by the following property:
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Let  $  E  ^ {*} $
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be the dual module of  $  E $,
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and let  $  \{ e _ {1}  ^ {*}, \dots, e _ {n}  ^ {*} \} $
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be the basis of  $  E  ^ {*} $
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dual to the initial basis of  $  E $:
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$  e _ {i}  ^ {*} ( e _ {i} ) = 1 $,
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$  e _ {i}  ^ {*} ( e _ {j} )= 0 $,
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$  i \neq j $.
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To each bilinear form  $  f $
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on  $  E $
 +
there correspond mappings  $  \phi _ {f} , \psi _ {f} : E \rightarrow E  ^ {*} $,
 +
defined by the equations
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034080/d03408029.png" /></td> </tr></table>
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$$
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\phi _ {f} ( x) ( y)  = f ( x, y) ,\ \
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\psi _ {f} ( x) ( y)  = f ( y, x) .
 +
$$
  
 +
If the form  $  f $
 +
is non-singular,  $  \phi _ {f} , \psi _ {f} $
 +
are isomorphisms, and vice versa. Here the basis  $  \{ c _ {1}, \dots, c _ {n} \} $
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dual to  $  \{ e _ {1}, \dots, e _ {n} \} $
 +
is distinguished by the following property:
  
 +
$$
 +
\psi _ {f} ( c _ {i} )  =  e _ {i}  ^ {*} \ \
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( i = 1, \dots, n) .
 +
$$
  
 
====Comments====
 
====Comments====
A bilinear form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034080/d03408030.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034080/d03408031.png" /> is non-degenerate (also called non-singular) if for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034080/d03408032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034080/d03408033.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034080/d03408034.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034080/d03408035.png" /> and for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034080/d03408036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034080/d03408037.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034080/d03408038.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034080/d03408039.png" />. Occasionally the terminology conjugate module (conjugate space) is used instead of dual module (dual space).
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A bilinear form $  f $
 +
on $  E $
 +
is non-degenerate (also called non-singular) if for all $  x \in E $,
 +
$  f ( x , y ) = 0 $
 +
for all $  y $
 +
implies $  x = 0 $
 +
and for all $  y \in E $,
 +
$  f ( x , y ) = 0 $
 +
for all $  x $
 +
implies $  y = 0 $.  
 +
Occasionally the terminology conjugate module (conjugate space) is used instead of dual module (dual space).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.M. Cohn,  "Algebra" , '''1''' , Wiley  (1982)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.M. Cohn,  "Algebra" , '''1''' , Wiley  (1982)</TD></TR></table>

Latest revision as of 19:52, 1 February 2022


to a basis $ \{ e _ {1}, \dots, e _ {n} \} $ of a module $ E $ with respect to a form $ f $

A basis $ \{ c _ {1}, \dots, c _ {n} \} $ of $ E $ such that

$$ f ( e _ {i} , c _ {i} ) = 1 ,\ f ( e _ {i} , c _ {j} ) = 0 , $$

$$ i \neq j ,\ 1 \leq i , j \leq n , $$

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

Let $ E ^ {*} $ be the dual module of $ E $, and let $ \{ e _ {1} ^ {*}, \dots, e _ {n} ^ {*} \} $ be the basis of $ E ^ {*} $ dual to the initial basis of $ E $: $ e _ {i} ^ {*} ( e _ {i} ) = 1 $, $ e _ {i} ^ {*} ( e _ {j} )= 0 $, $ i \neq j $. To each bilinear form $ f $ on $ E $ there correspond mappings $ \phi _ {f} , \psi _ {f} : E \rightarrow E ^ {*} $, defined by the equations

$$ \phi _ {f} ( x) ( y) = f ( x, y) ,\ \ \psi _ {f} ( x) ( y) = f ( y, x) . $$

If the form $ f $ is non-singular, $ \phi _ {f} , \psi _ {f} $ are isomorphisms, and vice versa. Here the basis $ \{ c _ {1}, \dots, c _ {n} \} $ dual to $ \{ e _ {1}, \dots, e _ {n} \} $ is distinguished by the following property:

$$ \psi _ {f} ( c _ {i} ) = e _ {i} ^ {*} \ \ ( i = 1, \dots, n) . $$

Comments

A bilinear form $ f $ on $ E $ is non-degenerate (also called non-singular) if for all $ x \in E $, $ f ( x , y ) = 0 $ for all $ y $ implies $ x = 0 $ and for all $ y \in E $, $ f ( x , y ) = 0 $ for all $ x $ implies $ y = 0 $. 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. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dual_basis&oldid=14785
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