one-to-one and invertible

Before looking at the complications caused by limiting processes, it is convenient to begin with the simpler functions, and to look for functions with simpler properties. Functions can be defined explicitly as well as implicitly, for example by equating polynomials in two variables to zero. Bearing in mind that polynomials have as many roots as their degree, the polynomials should be limited to the first degree to avoid multiple valuedness in the inverse function. Therefore the relation

d w z + c w + b z + a = 0 (18)

is reccommended. It can be rendered explicit in either direction,

$$-\frac{bz+a}{dz+c}$$ w = (19)

$$-\frac{cw+a}{dw+b}$$ z = (20)

Inspection, and particularly familiarity with projective transformations, suggests a matrix representation of these fractional linear transformations. To get such a representation, the complex variables should be represented as quotients, such as w = s/t, z = u/v. Then the numerator and denominator of the fractional linear transformation are subject to linear transformations for which the matrix notation is appropriate.

$$\left[ \begin{array}{c} s\\ t \end{array} \right] = \left[ ... ...y} \right] \left[ \begin{array}{c} u\\ v \end{array} \right]$$

Just because the complex variable is represented as a quotient (and observe that the members of the numerator-denominator pair are not the real and imaginary parts of the variable, but are themselves complex numbers) ambiguity exists both in the representation of the variable and in the matrix describing the transformation.

Since the matrix should correspond to an invertible transformation, its determinant should not vanish. Accordingly the matrix could be multiplied by a factor making the determinant unity. Such a choice which will later on prove to be consistent because the matrices of composite transformations multiply and so do the determinants.