The Klein View of Geometry 
Definition The Mobius group M(2) consists of all bilinear transformations of the form
where α, β, γ, δ are in C, with αδ  βγ ≠ 0.


This defines a transformation of E^{+} once we put




It is easier to use matrices to calculate composites and inverses in M(2). Suppose that m and n are the Mobius transformations m(z) = (αz+β)/(γz+δ), and n(z) = (α'z+β')/(γ'z+δ'), With the transformations m and n , we associate the matrices
Thus mon has matrix
It follows easily that the inverse of m has matrix M^{1},
In fact, since matrices A and λA give rise to the same transformation,
