affine transformation
Definition 1.
Let $\mathrm{(}{A}_{i}\mathrm{,}{f}_{i}\mathrm{)}$ be affine spaces^{} associated with a left (right) vector spaces^{} ${V}_{i}$ (over some division ring $D$), where $i\mathrm{=}\mathrm{1}\mathrm{,}\mathrm{2}$. An affine transformation^{} from ${A}_{\mathrm{1}}$ to ${A}_{\mathrm{2}}$ is a function $\alpha \mathrm{:}{A}_{\mathrm{1}}\mathrm{\to}{A}_{\mathrm{2}}$ such that there is a linear transformation $T\mathrm{:}{V}_{\mathrm{1}}\mathrm{\to}{V}_{\mathrm{2}}$ such that
$$T({f}_{1}(P,Q))={f}_{2}(\alpha (P),\alpha (Q))$$ 
for any $P\mathrm{,}Q\mathrm{\in}A$.
Note that $T$ is uniquely determined by $\alpha $, since ${f}_{1}$ is a function onto ${V}_{1}$. $T$ and is called the associated linear transformation of $\alpha $. Let us write $[\alpha ]$ the associated linear transformation of $\alpha $. Then the definition above can be illustrated by the following commutative diagram^{}:
$$\text{xymatrix}\mathrm{@}+=2cm{A}_{1}\times {A}_{1}\text{ar}{[r]}^{}{f}_{1}\text{ar}{[d]}_{(\alpha ,\alpha )}\mathrm{\&}{V}_{1}\text{ar}{[d]}^{[\alpha ]}{A}_{2}\times {A}_{2}\text{ar}{[r]}_{}{f}_{2}\mathrm{\&}{V}_{2}$$ 
Here’s an example of an affine transformation. Let $(A,f)$ be an affine space with $V$ the associated vector space. Fix $v\in V$. For each $P\in A$, let $\alpha (P)$ be the unique point in $A$ such that $f(P,\alpha (P))=v$. Then $\alpha :A\to A$ is a welldefined function. Furthermore, $f(\alpha (P),\alpha (Q))=v+f(\alpha (P),\alpha (Q))v=f(P,\alpha (P))+f(\alpha (P),\alpha (Q))+f(\alpha (Q),Q)=f(P,Q)={1}_{V}(f(P,Q))$. Thus $\alpha $ is affine, with $[\alpha ]={1}_{V}$.
An affine transformation $\alpha :{A}_{1}\to {A}_{2}$ is an affine isomorphism if there is an affine transformation $\beta :{A}_{2}\to {A}_{1}$ such that $\beta \circ \alpha ={1}_{{A}_{1}}$ and $\alpha \circ \beta ={1}_{{A}_{2}}$. Two affine spaces ${A}_{1}$ and ${A}_{2}$ are affinely isomorphic, or simply, isomorphic^{}, if there are affine isomorphism $\alpha :{A}_{1}\to {A}_{2}$.
Below are some basic properties of an affine transformation (see proofs here (http://planetmath.org/PropertiesOfAnAffineTransformation)):

1.
$\alpha $ is onto iff $[\alpha ]$ is.

2.
$\alpha $ is onetoone iff $[\alpha ]$ is.

3.
A bijective^{} affine transformation $\alpha $ is an affine isomorphism. In fact, $[{\alpha}^{1}]={[\alpha ]}^{1}$.

4.
Two affine spaces associated with the same vector space are isomorphic.
Because of the last property, it is often enough, in practice, to identify $V$ itself as the affine space associated with $V$, up to affine isomorphism, with the direction given by $f(v,w)=wv$. With this in mind, we may reformulate the definition of an affine transformation as a mapping $\alpha $ from one vector space $V$ to another, $W$, such that there is a linear transformation $T:V\to W$ such that
$$T(wv)=\alpha (w)\alpha (v).$$ 
By fixing $w\in V$, we get the following equation
$$\alpha (v)=T(v)+(\alpha (w)T(w)).$$ 
Definition 2.
Let $V$ and $W$ be left vector spaces over the same division ring $D$. An affine transformation is a mapping $\alpha \mathrm{:}\mathrm{:}V\mathrm{\to}W$ such that
$$\alpha (v)=T(v)+w,v\in V$$ 
for some linear transformation $T\mathrm{:}V\mathrm{\to}W$ and some vector $w\mathrm{\in}W$.
An affine property is a geometry^{} property that is preserved by an affine transformation. The following are affine properties of an affine transformation Let $A:V\to W$:

•
linearity. Given an affine subspace $S+v$ of $V$, then $A(S+v)=L(S+v)+w=L(S)+(L(v)+w)$ is an affine subspace of $W$.

•
incidence. Suppose $S+v\subseteq T+u$. Pick $x\in A(S+v)=L(S)+L(v)+w$, so $x=y+L(v)+w$ where $y\in L(S)$. Since $L$ is bijective, there is $z\in S$ such that $L(z)=y$. So $A(z+v)=L(z)+L(v)+w=x$. Since $z+v\in S+v$, $z+v=t+u$ for some $t\in T$, $x=A(z+v)=A(t+u)\in A(T+u)$. Therefore, $A(S+v)\subseteq A(T+u)$.

•
parallelism^{}. Given two parallel affine subspaces $S+a$ and $S+b$, then $A(S+a)=L(S)+(L(a)+w)$ and $A(S+b)=L(S)+(L(b)+w)$ are parallel.

•
coefficients of an affine combination. Given that $v$ is an affine combination of ${v}_{1},\mathrm{\dots},{v}_{n}$:
$$v={k}_{1}{v}_{1}+\mathrm{\cdots}+{k}_{n}{v}_{n},$$ where ${k}_{i}\in F\text{and}{k}_{1}+\mathrm{\cdots}+{k}_{n}=1$ are the corresponding coefficients. Then
$A(v)$ $=$ ${k}_{1}L({v}_{1})+\mathrm{\cdots}+{k}_{n}L({v}_{n})+w$ $=$ ${k}_{1}(L({v}_{1})+w)+\mathrm{\cdots}+{k}_{n}(L({v}_{n})+w)$ $=$ ${k}_{1}A({v}_{1})+\mathrm{\cdots}+{k}_{n}A({v}_{n})$ is the affine combination of $A({v}_{1}),\mathrm{\dots},A({v}_{n})$ with the same set of coefficients.
Special Affine Transformations

1.
translation. An affine transformation of the form $A(v)=v+w$ is called a translation. Every affine transformation can be decomposed as a product^{} of a linear transformation and a translation: $A(v)=L(v)+w=BC(v)$ where $C(v)=L(v)$ and $B(v)=v+w$. The order of composition^{} is important, since $BC\ne CB$. Geometrically, a translation moves a geometric figure along a straight line.

2.
dilation (map). If $L$ has a unique eigenvalue^{} $d\ne 0$ (that is, $L$ may be diagonalized as $dI$, the diagonal matrix^{} with nonzero diagonal^{} entries $=d\in F$), then the affine transformation $A(v)=L(v)$ is called a dilation. Note that a dilation may be written as the product of a vector with a scalar: $A(v)=dv$, which is why a dilation is also called a scaling. A dilation can be visualized as magnifying or shrinking a geometric figure.

3.
homothetic transformation. The composition of a dilation followed by a translation is called a homothetic transformation. It has the form $A(v)=dv+w$, $0\ne d\in F$.

4.
Euclidean transformation. In the case when both $V$ and $W$ are Euclidean vector spaces, if the associated linear transformation is orthogonal^{}, then the affine transformation is called a Euclidean transformation.
Remarks

1.
When $V=W$, the set of affine maps $V\to V$, with function composition as the product, becomes a group, and is denoted by $\mathrm{IGL}(V)$. The multiplicative identity^{} is the identity map^{}. If $A(v)=L(v)+w$, then ${A}^{1}(v)={L}^{1}(v){L}^{1}(w)$. IGL is short for of $V$. Translations, dilations, and homothetic transformations all form subgroups^{} of $\mathrm{IGL}(V)$. If $T$ is the group of translations, $D$ the group of dilations, and $H$ the group of homothetic transformations, then $T$ is a normal subgroup^{} of $T$. Also, $\mathrm{Aut}(T)$ and $\mathrm{Aut}(D)$ are abelian groups^{} (remember: $F$ is assumed to be a field).

2.
One can more generally define an affine transformation to be an orderpreserving bijection between two affine geometries. It can be shown that this definition coincides with the above one if the underlying field admits no nontrivial automorphisms^{}. When the two affine geometries are the same, the bijective affine transformation is called an affinity.

3.
Another way to generalize an affine transformation is to remove the restriction^{} on the invertibility of the linear transformation $L$. In this respect, the set $A(V,W)$ of affine transformations from $V$ to $W$ has a natural vector space structure^{}. It is easy to see that the set $L(V,W)$ of linear transformations from $V$ to $W$ forms a subspace^{} of $A(V,W)$.
Title  affine transformation 
Canonical name  AffineTransformation 
Date of creation  20130322 14:46:08 
Last modified on  20130322 14:46:08 
Owner  matte (1858) 
Last modified by  matte (1858) 
Numerical id  37 
Author  matte (1858) 
Entry type  Definition 
Classification  msc 51A10 
Classification  msc 51A15 
Classification  msc 15A04 
Synonym  scaling 
Related topic  LinearTransformation 
Related topic  AffineSpace 
Related topic  ComplexLine 
Related topic  AffineCombination 
Related topic  AffineGeometry 
Related topic  Collineation^{} 
Defines  IGL 
Defines  translation 
Defines  dilation 
Defines  dilation map 
Defines  homothetic transformation 
Defines  affine property 
Defines  affine isomorphism 
Defines  associated linear transformation 
Defines  affinely isomorphic 
Defines  affinity 