Editing Affine transformation (section)

==Definition==
Let {{mvar|X}} be an affine space over a [[field (mathematics)|field]] {{mvar|k}}, and {{mvar|V}} be its associated vector space. An '''affine transformation''' is a [[bijection]] {{mvar|f}} from {{mvar|X}} onto itself that is an [[affine space#Affine map|affine map]]; this means that  a [[linear map]] {{mvar|g}} from {{mvar|V}} to {{mvar|V}} is well defined by the equation <math>g(y-x) =f(y)-f(x);</math> here, as usual, the subtraction of two points denotes the [[free vector]] from the second point to the first one, and "[[well-defined]]" means that <math>y-x= y'-x'</math> implies that <math>f(y)-f(x)=f(y')-f(x').</math>

If the dimension of {{mvar|X}} is at least two, a  ''semiaffine transformation'' {{mvar|f}} of {{mvar|X}} is a [[bijection]] from {{mvar|X}} onto itself satisfying:{{sfn|Snapper|Troyer|1989|p=65}}
#For every {{math|''d''}}-dimensional [[affine subspace]] {{mvar|S}} of {{mvar|X}}, then {{math|''f'' (''S'')}} is also a {{math|''d''}}-dimensional affine subspace of {{mvar|X}}.
#If {{mvar|S}} and {{mvar|T}} are parallel affine subspaces of {{mvar|X}}, then {{math|''f'' (''S'')}} and  {{math|''f'' (''T'')}} are parallel.
These two conditions are satisfied by affine transformations, and express what is precisely meant by the expression that "{{mvar|f}} preserves parallelism".

These conditions are not independent as the second follows from the first.{{sfn|Snapper|Troyer|1989|p=66}} Furthermore, if the field {{mvar|k}} has at least three elements, the first condition can be simplified to: {{mvar|f}} is a [[collineation]], that is, it maps lines to lines.{{sfn|Snapper|Troyer|1989|p=69}}