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Recall from an earlier section that a Geometry consists of a set *S* (usually **R**^{n} for us) together with a group *G* of transformations acting on *S*.

We now examine some natural groups which are *bigger* than the Euclidean group. Although the geometry we get is not Euclidean, they are not called *non-Euclidean* since this term is reserved for something else.

**Definition**

An **affine transformation** or **affinity** of **R**^{n} is one of the form *T*_{a} ∘ *L* with *T*_{a} a translation and *L* ∈ *GL*(*n*, **R**).

The group of all such transformations is called the **Affine group** and is written *A*(**R**^{n}).

**Examples**

- (
*x*,*y*) ↦ (1 + 2*x*, 1 + 2*y*)

- (
*x*,*y*) ↦ (1 +*x*+*y*, 2 +*y*)

Note that affine transformations do

They do preserve some geometric properties.

**Collinearity***If A, B and C are collinear, so are their images under any affine map.*More generally, we have:

**Definition**A translation of a linear subspace of

**R**^{n}is called an**affine subspace**.For example, any line or plane in

**R**^{3}is an affine subspace.**Theorem**

*Affine transformations map affine subspaces to affine subspaces.***Proof**

This follows from the fact that linear maps map linear subspaces to linear subspaces.

**Parallelism****Theorem**

*Parallel lines are mapped to parallel lines.***Proof**

Two parallel lines are lines in an affine plane which do not meet. Since affine transformations preserve planes and incidence, their images lie in an affine plane and do not meet. Hence they are parallel.

**Ratios****Theorem**

*Ratios of lengths of intervals on any line are preserved.***Proof**

This follows because such ratios are preserved by linear maps and by translations.

In fact ratios of lengths on pairs of parallel lines are preserved.

The property of*intermediacy*(one point being*between*two others on a line) is also preserved.

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