Let $\widehat{\mathbb{C}}=\mathbb{C}\cup\{\infty\}$ be the extended complex plane. Let $\operatorname{Mob}(\widehat{\mathbb{C}})$ denote the set of all maps $T:\widehat{\mathbb{C}}\to\widehat{\mathbb{C}}$ of the form

\begin{align*} T(z)=\frac{az+b}{cz+d} \end{align*}

for some $a,b,c,d\in\mathbb{C}$ with $ad-bc\ne 0$, where the associated matrix has first row $(a,b)$ and second row $(c,d)$, with the usual extended-value conventions at $\infty$ and at the finite pole when $c\ne 0$. Then $\operatorname{Mob}(\widehat{\mathbb{C}})$ is a group under composition. Moreover, the map from $GL(2,\mathbb{C})$ to $\operatorname{Mob}(\widehat{\mathbb{C}})$ sending the matrix whose first row is $(a,b)$ and whose second row is $(c,d)$ to the corresponding transformation has kernel equal to the subgroup of nonzero scalar matrices, and hence induces a group isomorphism

\begin{align*} PGL(2,\mathbb{C})=GL(2,\mathbb{C})/\mathbb{C}^{\times}\cong \operatorname{Mob}(\widehat{\mathbb{C}}), \end{align*}

where $\mathbb{C}^{\times}$ is identified with the subgroup of nonzero scalar matrices.