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:

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. 1.

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

2. 2.

   $\alpha$ is one-to-one iff $[\alpha]$ is.

3. 3.

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

4. 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

By fixing $w\in V$, we get the following equation

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.