Editing Affine transformation (section)

==Structure==
By the definition of an affine space, {{mvar|V}} acts on {{mvar|X}}, so that, for every pair <math>(x, \mathbf{v})</math> in {{math|''X'' × ''V''}} there is associated a point {{mvar|y}} in {{mvar|X}}. We can denote this action by  <math>\vec{v}(x) = y</math>. Here we use the convention that  <math>\vec{v} = \textbf{v}</math> are two interchangeable notations for an element of {{mvar|V}}.  By fixing a point {{mvar|c}} in {{mvar|X}} one can define a function {{math|''m''<sub>''c''</sub> : ''X'' → ''V''}} by {{math|1=''m''<sub>''c''</sub>(''x'') = {{vec|''cx''}}}}. For any {{mvar|c}}, this function is one-to-one, and so, has an inverse function {{math|''m''<sub>''c''</sub><sup>−1</sup> : ''V'' → ''X''}} given by <math>m_c^{-1}(\textbf{v})=\vec{v}(c)</math>. These functions can be used to turn {{mvar|X}} into a vector space (with respect to the point {{mvar|c}}) by defining:{{sfn|Snapper|Troyer|1989|p=59}}
:* <math>x + y = m_c^{-1}\left(m_c(x) + m_c(y)\right),\text{ for all } x,y \text{ in } X,</math> and
:* <math>rx = m_c^{-1}\left(r m_c(x)\right), \text{ for all } r \text{ in } k \text{ and } x \text{ in } X.</math>
This vector space has origin {{mvar|c}} and formally needs to be distinguished from the affine space {{mvar|X}}, but common practice is to denote it by the same symbol and mention that it is a vector space ''after'' an origin has been specified. This identification permits points to be viewed as vectors and vice versa.

For any [[linear transformation]] {{mvar|λ}} of {{mvar|V}}, we can define the function {{math|''L''(''c'', ''λ'') : ''X'' → ''X''}} by
:<math>L(c, \lambda)(x) = m_c^{-1}\left(\lambda (m_c (x))\right) = c + \lambda (\vec{cx}).</math>

Then {{math|''L''(''c'', ''λ'')}} is an affine transformation of {{mvar|X}} which leaves the point {{mvar|c}} fixed.{{sfn|Snapper|Troyer|1989|p=76,87}} It is a linear transformation of {{mvar|X}}, viewed as a vector space with origin {{mvar|c}}.

Let {{mvar|σ}} be any affine transformation of {{mvar|X}}. Pick a point {{mvar|c}} in {{mvar|X}} and consider the translation of {{mvar|X}} by the vector <math>\bold{w} = \overrightarrow{c \sigma (c)}</math>, denoted by {{math|''T''<sub>'''w'''</sub>}}. Translations are affine transformations and the composition of affine transformations is an affine transformation. For this choice of {{mvar|c}}, there exists a unique linear transformation {{mvar|λ}} of {{mvar|V}} such that{{sfn|Snapper|Troyer|1989|p=86}} 
:<math>\sigma (x) = T_{\bold{w}} \left( L(c, \lambda)(x) \right).</math>
That is, an arbitrary affine transformation of {{mvar|X}} is the composition of a linear transformation of {{mvar|X}} (viewed as a vector space) and a translation of {{mvar|X}}.

This representation of affine transformations is often taken as the definition of an affine transformation (with the choice of origin being implicit).{{sfn|Wan|1993|pp=19-20}}{{sfn|Klein|1948|p=70}}{{sfn|Brannan|Esplen|Gray|1999|p=53}}