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In the [[mathematics|mathematical]] study of [[transformation group]]s, the '''special affine group''' is the [[group (mathematics)|group]] of [[affine transformation]]s of a fixed [[affine space]] which preserve [[volume]]. This is the affine analogue of the [[special linear group]].
 
On '''R'''<sup>n</sup>, every special affine transformation has the form
 
:<math>\mathbf{x}\mapsto A\mathbf{x}+\mathbf{b}</math>
 
where ''A'' is an ''n''&times;''n'' [[matrix (mathematics)|matrix]] of [[determinant]] one, and '''b''' is a fixed vector in '''R'''<sup>n</sup>.  The special affine group is a [[subgroup]] of the [[affine group]].
 
{{geometry-stub}}
[[Category:Lie groups]]

Latest revision as of 07:45, 18 March 2013

In the mathematical study of transformation groups, the special affine group is the group of affine transformations of a fixed affine space which preserve volume. This is the affine analogue of the special linear group.

On Rn, every special affine transformation has the form

xAx+b

where A is an n×n matrix of determinant one, and b is a fixed vector in Rn. The special affine group is a subgroup of the affine group.



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