Editing Projective linear group (section)

=== Fractional linear transformations ===
{{details|Möbius transformation#Projective matrix representations}}
As for [[Möbius transformation]]s, the group {{nowrap|PGL(2, ''K'')}} can be interpreted as [[fractional linear transformation]]s with coefficients in ''K''. Points in the projective line over ''K'' correspond to pairs from ''K''<sup>2</sup>, with two pairs being equivalent when they are proportional. When the second coordinate is non-zero, a point can be represented by {{nowrap|[''z'', 1]}}. Then when {{nowrap|''ad'' − ''bc'' ≠ 0}}, the action of {{nowrap|PGL(2, ''K'')}} is by linear transformation:
: <math>[z,\ 1]\begin{pmatrix} a & c \\ b & d \end{pmatrix} \ = \ [az + b,\ cz + d] \ = \ \left [\frac{a z + b}{c z + d},\ 1\right ].</math>
In this way successive transformations can be written as right multiplication by such matrices, and [[matrix multiplication]] can be used for the group product in {{nowrap|PGL(2, ''K'')}}.