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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+1 = 39 n = 0 |
| + | $#C+1 = 39 : ~/encyclopedia/old_files/data/D034/D.0304080 Dual basis |
| + | Automatically converted into TeX, above some diagnostics. |
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| + | if TeX found to be correct. |
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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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− | <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>
| + | ''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>
| + | 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" />.
| + | $$ |
| + | f ( e _ {i} , c _ {i} ) = 1 ,\ f ( e _ {i} , c _ {j} ) = 0 , |
| + | $$ |
| | | |
− | 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
| + | $$ |
| + | i \neq j ,\ 1 \leq i , j \leq n , |
| + | $$ |
| | | |
− | <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>
| + | 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 $. |
| | | |
− | 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:
| + | 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 |
| | | |
− | <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>
| + | $$ |
| + | \phi _ {f} ( x) ( y) = f ( x, y) ,\ \ |
| + | \psi _ {f} ( x) ( y) = f ( y, x) . |
| + | $$ |
| | | |
| + | 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==== | | ====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). | + | 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> |
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:
$$
\psi _ {f} ( c _ {i} ) = e _ {i} ^ {*} \ \
( i = 1 \dots n) .
$$
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) |