These mappings map lines and circles to either lines or circles.
Definition 17.4.1 A fractional linear transformation is a function of the form
 (17.2) 
where ad − bc≠0.
Note that if c = 0, this reduces to a linear transformation


The next lemma is the key to understanding fractional linear transformations.
Lemma 17.4.2 The fractional linear transformation, 17.2 can be written as a finite composition of dilations, inversions, and translations.
Proof: Let

and

in the case where c≠0. Then f

Here is why.

Now consider

Finally, consider

In case that c = 0, f
This lemma implies the following corollary.
Corollary 17.4.3 Fractional linear transformations map circles and lines to circles or lines.
Proof: It is obvious that dilations and translations map circles to circles and lines to lines. What of inversions? If inversions have this property, the above lemma implies a general fractional linear transformation has this property as well.
Note that all circles and lines may be put in the form

where α = 1 gives a circle centered at
 (17.3) 
To see this let β = β_{1} + iβ_{2} where β_{1} ≡−a and β_{2} ≡ b. Note that even if α is not 0 or 1 the expression still corresponds to either a circle or a line because you can divide by α if α≠0. Now I verify that replacing z with

and so w also satisfies a relation like 17.3. One simply switches α with γ and β with β. Note the situation is slightly different than with dilations and translations. In the case of an inversion, a circle becomes either a line or a circle and similarly, a line becomes either a circle or a line. ■
The next example is quite important.
Example 17.4.4 Consider the fractional linear transformation, w =
First consider what this mapping does to the points of the form z = x + i0. Substituting into the expression for w,

a point on the unit circle. Thus this transformation maps the real axis to the unit circle.
The upper half plane is composed of points of the form x + iy where y > 0. Substituting in to the transformation,

which is seen to be a point on the interior of the unit disk because
One might wonder whether the mapping is one to one and onto. The mapping is clearly one to one because it has an inverse, z = −i