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 well-defined 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):

1. $\alpha$ is onto iff $[\alpha]$ is.
2. $\alpha$ is one-to-one 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 $f(v,w)=w-v$ . 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 $$T(w-v)=\alpha(w)-\alpha(v).$$ By fixing $w\in V$ , we get the following equation $$\alpha(v)=T(v)+(\alpha(w)-T(w)).$$

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 $v_1,\ldots,v_n$ : $$v=k_1v_1+\cdots +k_nv_n,$$ where $k_i\in F\mbox{ and }k_1+\cdots+k_n=1$ are the corresponding coefficients. Then \begin{eqnarray*} A(v) &=& k_1L(v_1)+\cdots+k_nL(v_n)+w \\ &=& k_1(L(v_1)+w)+\cdots+k_n(L(v_n)+w) \\ &=& k_1A(v_1)+\cdots+k_nA(v_n) \end{eqnarray*}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 non-zero 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 order-preserving bijection between two affine geometries. It can be shown that this definition coincides with the above one if the underlying field admits no non-trivial automorphisms.
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 $A(V,W)$ .