Franklin's Notes

Mobius transformation

If $\mathbb k$ is a field and $\hat{\mathbb k}=\mathbb k\cup{\infty}$, then a mobius transformation is a function of the form where $ad-bc\ne 0$, defined with domain and codomain $\hat{\mathbb k}$ such that $f(-d/c)=\infty$ and $f(\infty)=a/c$. Most often, mobius transformations are considered in complex projective space $\hat{\mathbb C}$, where they are the simplest kind of rational function (i.e. a rational function of order $1$) and have remarkable geometric properties.

Proposition 1. For any field $\mathbb k$, a mobius transformation is a bijection $\hat{\mathbb k}\to\hat{\mathbb k}$.

Proof. To see why $f$ is bijective, it suffices to express it as a composition of bijections on $\hat{\mathbb k}$. If $c=0$, then $f$ is a linear function, making it bijective. On the other hand, if $c\ne 0$, we may write so that $f$ is the composition of the following three maps: Notice that $b-dac^{-1}\ne 0$, since $ad-bc\ne 0$ as part of the definition of a mobius transformation. This means that $f$ is the composition of two linear functions with leading coefficient $\ne 0$ and an inversion. Linear function with nonzero leading coefficient are bijective in $\hat{\mathbb k}$, since they are bijective on $\mathbb k$ and send $\infty\mapsto\infty$, and inversion is also bijective since it sends each nonzero element of $\mathbb k$ to its multiplicative inverse and $0\leftrightarrow \infty$. Thus, $f$ is bijective. $\blacksquare$



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