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Difference between revisions of "Unimodular transformation"

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A [[Linear transformation|linear transformation]] of a finite-dimensional [[Vector space|vector space]] whose matrix has determinant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095380/u0953801.png" />.
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A [[Linear transformation|linear transformation]] of a finite-dimensional [[Vector space|vector space]] whose matrix has determinant  $  \pm  1 $.
  
 
====Comments====
 
====Comments====
The name  "unimodular transformation"  is often restricted to mean a linear transformation with determinant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095380/u0953802.png" />. In the context of a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095380/u0953803.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095380/u0953804.png" /> which is the quotient field of an integral domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095380/u0953805.png" />, with a fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095380/u0953806.png" />-basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095380/u0953807.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095380/u0953808.png" />, a linear transformation is called unimodular if its matrix with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095380/u0953809.png" /> has entries in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095380/u09538010.png" /> and determinant a unit in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095380/u09538011.png" />. Under each of these definitions the unimodular transformations form a group. In the case of linear transformations with determinant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095380/u09538012.png" /> one often calls this the [[Unimodular group|unimodular group]], or, more commonly nowadays, the [[Special linear group|special linear group]].
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The name  "unimodular transformation"  is often restricted to mean a linear transformation with determinant $  1 $.  
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In the context of a vector space $  V $
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over a field $  k $
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which is the quotient field of an integral domain $  D $,  
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with a fixed $  k $-
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basis $  a _ {1} \dots a _ {n} $
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in $  V $,  
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a linear transformation is called unimodular if its matrix with respect to $  a _ {1} \dots a _ {n} $
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has entries in $  D $
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and determinant a unit in $  D $.  
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Under each of these definitions the unimodular transformations form a group. In the case of linear transformations with determinant $  1 $
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one often calls this the [[Unimodular group|unimodular group]], or, more commonly nowadays, the [[Special linear group|special linear group]].

Latest revision as of 08:27, 6 June 2020


A linear transformation of a finite-dimensional vector space whose matrix has determinant $ \pm 1 $.

Comments

The name "unimodular transformation" is often restricted to mean a linear transformation with determinant $ 1 $. In the context of a vector space $ V $ over a field $ k $ which is the quotient field of an integral domain $ D $, with a fixed $ k $- basis $ a _ {1} \dots a _ {n} $ in $ V $, a linear transformation is called unimodular if its matrix with respect to $ a _ {1} \dots a _ {n} $ has entries in $ D $ and determinant a unit in $ D $. Under each of these definitions the unimodular transformations form a group. In the case of linear transformations with determinant $ 1 $ one often calls this the unimodular group, or, more commonly nowadays, the special linear group.

How to Cite This Entry:
Unimodular transformation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Unimodular_transformation&oldid=49080
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article