Editing Affine connection (section)

====Definition of an affine space====

Informally, an '''[[affine space]]''' is a [[vector space]] without a fixed choice of [[origin (mathematics)|origin]].  It describes the geometry of [[point (mathematics)|points]] and [[Vector (geometric)|free vectors]] in space.  As a consequence of the lack of origin, points in affine space cannot be added together as this requires a choice of origin with which to form the parallelogram law for vector addition.  However, a vector {{mvar|v}} may be added to a point {{mvar|p}} by placing the initial point of the vector at {{mvar|p}} and then transporting {{mvar|p}} to the terminal point. The operation thus described {{math|''p'' → ''p'' + ''v''}} is the '''translation''' of {{mvar|p}} along {{mvar|v}}. In technical terms, affine {{mvar|n}}-space is a set {{math|'''A'''<sup>''n''</sup>}} equipped with a [[Group action (mathematics)|free transitive action]] of the vector group {{math|'''R'''<sup>''n''</sup>}} on it through this operation of translation of points: {{math|'''A'''<sup>''n''</sup>}} is thus a [[principal homogeneous space]] for the vector group {{math|'''R'''<sup>''n''</sup>}}.

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The [[general linear group]] {{math|GL(''n'')}} is the [[transformation group|group of transformations]] of {{math|'''R'''<sup>''n''</sup>}} which preserve the ''linear structure'' of {{math|'''R'''<sup>''n''</sup>}} in the sense that {{math|''T''(''av'' + ''bw'') {{=}} ''aT''(''v'') + ''bT''(''w'')}}.  By analogy, the '''[[affine group]]''' {{math|Aff(''n'')}} is the group of transformations of {{math|'''A'''<sup>''n''</sup>}} preserving the ''affine structure''.  Thus {{math|''φ'' ∈ Aff(''n'')}} must ''preserve translations'' in the sense that

:<math>\varphi(p+v)=\varphi(p)+T(v)</math>

where {{mvar|T}} is a general linear transformation. The map sending {{math|''φ'' ∈ Aff(''n'')}} to {{math|''T'' ∈ GL(''n'')}} is a [[group homomorphism]].  Its [[kernel (algebra)|kernel]] is the group of translations {{math|'''R'''<sup>''n''</sup>}}. The [[Group action (mathematics)#Orbits and stabilizers|stabilizer]] of any point {{mvar|p}} in {{mvar|A}} can thus be identified with {{math|GL(''n'')}} using this projection: this realises the affine group as a [[semidirect product]] of {{math|GL(''n'')}} and {{math|'''R'''<sup>''n''</sup>}}, and affine space as the [[homogeneous space]] {{math|Aff(''n'')/GL(''n'')}}.