Affine transformation
Structure

By the definition of an affine space, V acts on X, so that, for every pair (x, v) {\displaystyle (x,\mathbf {v})} in X × V there is associated a point y in X. We can denote this action by v → (x) = y {\displaystyle {\vec {v}}(x)=y}. Here we use the convention that v → = v {\displaystyle {\vec {v}}={\textbf {v}}} are two interchangeable notations for an element of V. By fixing a point c in X one can define a function m : X → V by m (x) = cx →. For any c, this function is one-to-one, and so, has an inverse function m : V → X given by m c − 1 (v) = v → (c) {\displaystyle m_{c}^{-1}({\textbf {v}})={\vec {v}}(c)}. These functions can be used to turn X into a vector space (with respect to the point c) by defining:

This vector space has origin c and formally needs to be distinguished from the affine space 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 λ of V, we can define the function L (c, λ) : X → X by

Then L (c, λ) is an affine transformation of X which leaves the point c fixed. It is a linear transformation of X, viewed as a vector space with origin c.

Let σ be any affine transformation of X. Pick a point c in X and consider the translation of X by the vector w = c σ (c) → {\displaystyle {\mathbf {w}}={\overrightarrow {c\sigma (c)}}}, denoted by T. Translations are affine transformations and the composition of affine transformations is an affine transformation. For this choice of c, there exists a unique linear transformation λ of V such that

That is, an arbitrary affine transformation of X is the composition of a linear transformation of X (viewed as a vector space) and a translation of X.

This representation of affine transformations is often taken as the definition of an affine transformation (with the choice of origin being implicit).
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