Given a vector space $V$ the projective linear group or projective general linear group is
$$ \mathbb{P}GL(V)=GL(V)/Z(V) $$where $GL(V)$ is the general linear group of $V$ and $Z(V)$ is the subgroup of all nonzero scalar transformations of $V$ (which is the center of this group). Their elements are called homographies.
It acts on the projective space associated to $V$, $\mathbb{P}(V)$, in the following way. Given $[T]\in \mathbb{P}GL(V)$ we have
$$ [T][Z]=[T(Z)] $$for $[Z]\in \mathbb{P}(V)$.
In case $V=\mathbb{C}^2$ we have the Moebius transformations.
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Author of the notes: Antonio J. Pan-Collantes
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