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affine transformation
Definition 1.
Let $(A_{i},f_{i})$ be affine spaces associated with a left (right) vector spaces $V_{i}$ (over some division ring $D$), where $i=1,2$. An affine transformation from $A_{1}$ to $A_{2}$ is a function $\alpha:A_{1}\to A_{2}$ such that there is a linear transformation $T:V_{1}\to V_{2}$ such that
$T(f_{1}(P,Q))=f_{2}(\alpha(P),\alpha(Q))$ 
for any $P,Q\in A$.
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:
$\xymatrix@+=2cm{A_{1}\times A_{1}\ar[r]^{}{f_{1}}\ar[d]_{{(\alpha,\alpha)}}&V% _{1}\ar[d]^{{[\alpha]}}\\ A_{2}\times A_{2}\ar[r]_{}{f_{2}}&V_{2}}$ 
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):
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 $\alpha:\colon V\to W$ such that
$\alpha(v)=T(v)+w,\quad v\in V$ 
for some linear transformation $T\colon V\to W$ and some vector $w\in W$.
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 $v_{1},\ldots,v_{n}$:
$v=k_{1}v_{1}+\cdots+k_{n}v_{n},$ where $k_{i}\in F\mbox{ and }k_{1}+\cdots+k_{n}=1$ are the corresponding coefficients. Then
$\displaystyle A(v)$ $\displaystyle=$ $\displaystyle k_{1}L(v_{1})+\cdots+k_{n}L(v_{n})+w$ $\displaystyle=$ $\displaystyle k_{1}(L(v_{1})+w)+\cdots+k_{n}(L(v_{n})+w)$ $\displaystyle=$ $\displaystyle k_{1}A(v_{1})+\cdots+k_{n}A(v_{n})$ is the affine combination of $A(v_{1}),\ldots,A(v_{n})$ 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 $BC\neq CB$. Geometrically, a translation moves a geometric figure along a straight line.
2. dilation (map). If $L$ has a unique eigenvalue $d\neq 0$ (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 $A(v)=dv+w$, $0\neq d\in F$.
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 ${\rm IGL}(V)$. 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 Inhomogenous General Linear group of $V$. Translations, dilations, and homothetic transformations all form subgroups of ${\rm IGL}(V)$. 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, $\operatorname{Aut}(T)$ and $\operatorname{Aut}(D)$ 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 $A(V,W)$.
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Comments
affine transformation requirement
I think the linear transformation in the definition of affine transformation is required to be invertible. Correct?
Re: affine transformation requirement
It probably depends on the context.
In geometry, if one defines an affine map as
a map conserving straight lines, then L must
be invertible. For example, A(v)=w, maps
everything to a point. Also, in most practical
applications, L is probably invertible. Like
rotation about a point, etc.
I took this definition from the top of my head.
However, you have editing rights so please feel free
to make modifications.
Oh.. in Java, an affine map does not need to be
invertible :)
merganser.math.gvsu.edu/david/reed03/notes/chap4.pdf
transformation of convex polygons
Hello,
If a point P is inside a convex polygon Q and f is an affine transformation, will f(P) be inside f(Q)? i think the answer is yes, but why?
Re: transformation of convex polygons
Let P= sum_i {y_i v_i}, where the v_i are the vertices of Q, and the y_i are positive reals which sum to 1.
Let f be given by fx=Ax+w, for some matrix A and vector w.
Then
fP= AP + w
=A sum_i {y_i v_i} + w
= sum_i {y_i Av_i} + sum_i {y_i w}
= sum_i {y_i(Av_i +w)}
= sum_i {y_i fv_i}
Hence fP is in the convex hull of the f v_i. Therefore fP is in fQ.
minimal number of pts. to determine that a transform is affi...
I have the following question:
Let $V$ be a finitedimensional inner product space over a field $K$ with $\dim(V) = n$. Consider a finite set of points $v_1, \dots , v_m \in V$. Let $F:V\to V$. What is the lowest $m$ such that
if for all $i,j\in\{1, \dots, m\}$
$F(v_i)  F(v_j) = v_i  v_j$
then for all $v,w\in V$
$F(v)  F(w) = v  w$
???
I feel as though the answer is that the minimum such $m$ is $n+1$ based in Euclidean space intuition and I also feel that in this case the points should be affinely independent. Also, does one need to restrict the field $K$ in any way?
Re: minimal number of pts. to determine that a transform is ...
Sorry, I do not think this question is wellposed, please disregard it!