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A linear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093960/t0939601.png" /> of a (right) vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093960/t0939602.png" /> over a skew-field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093960/t0939603.png" /> with the properties
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{{TEX|done}}
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{{MSC|20}}
  
<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/t/t093/t093960/t0939604.png" /></td> </tr></table>
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A ''transvection'' is
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a linear mapping $f$ of a (right) vector space $V$ over a skew-field $K$ with the properties
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093960/t0939605.png" /> is the identity linear transformation. A transvection can be represented in the form
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$$\def\rk{\textrm{rk}\;}\rk(f-E) = 1\quad\textrm{and}\quad \textrm{Im}(f - E)\subseteq \ker(f-E),$$
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where $E$ is the identity linear transformation. A transvection can be represented in the form
  
<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/t/t093/t093960/t0939606.png" /></td> </tr></table>
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$$\def\a{\alpha} f(x) = x+a\a(x),$$
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where $a\in V$, $\a\in V^*$ and $\a(a) = 0$.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093960/t0939607.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093960/t0939608.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093960/t0939609.png" />.
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The transvections of a vector space $V$ generate the special linear, or unimodular, group $\def\SL{\textrm{SL}}\SL(V)$. It coincides with the commutator subgroup of $\def\GL{\textrm{GL}}\GL(V)$, with the exception of the cases when $\dim V = 1$ or $\dim V = 2$ and $K$ is the field of two elements. If $K$ is a field, then $\SL(V)$ is the group of matrices with determinant 1. In the general case (provided that $\dim V \ne 1$), $\SL(V)$ is the kernel of the epimorphism
  
The transvections of a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093960/t09396010.png" /> generate the special linear, or unimodular, group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093960/t09396011.png" />. It coincides with the commutator subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093960/t09396012.png" />, with the exception of the cases when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093960/t09396013.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093960/t09396014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093960/t09396015.png" /> is the field of two elements. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093960/t09396016.png" /> is a field, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093960/t09396017.png" /> is the group of matrices with determinant 1. In the general case (provided that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093960/t09396018.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093960/t09396019.png" /> is the kernel of the epimorphism
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$$\GL(V) \to K^*/[K^*,K^*],$$
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which is called the Dieudonné determinant (cf.
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[[Determinant|Determinant]]).
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====References====
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{|
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|-
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|valign="top"|{{Ref|Di}}||valign="top"|  J.A. Dieudonné,   "La géométrie des groupes classiques", Springer  (1955)     {{MR|0072144}}  {{ZBL|0067.26104}}
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|-
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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/t/t093/t093960/t09396020.png" /></td> </tr></table>
 
  
which is called the Dieudonné determinant (cf. [[Determinant|Determinant]]).
 
  
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.A. Dieudonné,  "La géométrie des groups classiques" , Springer  (1955)</TD></TR></table>
 
  
  
  
 
====Comments====
 
====Comments====
In the projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093960/t09396021.png" />, whose points are the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093960/t09396022.png" />-dimensional subspaces of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093960/t09396023.png" />, a transvection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093960/t09396024.png" /> as above induces a (projective) transvection with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093960/t09396025.png" /> as centre and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093960/t09396026.png" /> as axis. If one takes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093960/t09396027.png" /> to be a hyperplane at infinity in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093960/t09396028.png" />, such a transvection induces a translation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093960/t09396029.png" /> in the remaining affine space (interpreted as a linear space). See also [[Shear|Shear]].
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In the projective space $\P(V)$, whose points are the $1$-dimensional subspaces of $V$, a transvection $f$ as above induces a (projective) transvection with $aK$ as centre and $\ker(f-E)$ as axis. If one takes $\ker(f-E)$ to be a hyperplane at infinity in $\P(V)$, such a transvection induces a translation $x\mapsto x+b$ in the remaining affine space (interpreted as a linear space). See also
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[[Shear|Shear]].

Revision as of 18:43, 1 March 2012

2020 Mathematics Subject Classification: Primary: 20-XX [MSN][ZBL]

A transvection is a linear mapping $f$ of a (right) vector space $V$ over a skew-field $K$ with the properties

$$\def\rk{\textrm{rk}\;}\rk(f-E) = 1\quad\textrm{and}\quad \textrm{Im}(f - E)\subseteq \ker(f-E),$$ where $E$ is the identity linear transformation. A transvection can be represented in the form

$$\def\a{\alpha} f(x) = x+a\a(x),$$ where $a\in V$, $\a\in V^*$ and $\a(a) = 0$.

The transvections of a vector space $V$ generate the special linear, or unimodular, group $\def\SL{\textrm{SL}}\SL(V)$. It coincides with the commutator subgroup of $\def\GL{\textrm{GL}}\GL(V)$, with the exception of the cases when $\dim V = 1$ or $\dim V = 2$ and $K$ is the field of two elements. If $K$ is a field, then $\SL(V)$ is the group of matrices with determinant 1. In the general case (provided that $\dim V \ne 1$), $\SL(V)$ is the kernel of the epimorphism

$$\GL(V) \to K^*/[K^*,K^*],$$ which is called the Dieudonné determinant (cf. Determinant).

References

[Di] J.A. Dieudonné, "La géométrie des groupes classiques", Springer (1955) MR0072144 Zbl 0067.26104




Comments

In the projective space $\P(V)$, whose points are the $1$-dimensional subspaces of $V$, a transvection $f$ as above induces a (projective) transvection with $aK$ as centre and $\ker(f-E)$ as axis. If one takes $\ker(f-E)$ to be a hyperplane at infinity in $\P(V)$, such a transvection induces a translation $x\mapsto x+b$ in the remaining affine space (interpreted as a linear space). See also Shear.

How to Cite This Entry:
Transvection. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Transvection&oldid=21399
This article was adapted from an original article by E.B. Vinberg (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article