Projective linear group

Given a vector space $V$ the projective linear group or projective general linear group is

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.

In the language of this vault, these homographies are projectivities acting on the corresponding projective space $\mathbb{P}(V)$.

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

for $[Z]\in \mathbb{P}(V)$.

In case $V={\mathbb{C}}^2$ we have the Moebius transformations.

See also:

• projective line
• cross-ratio
• Fundamental theorem of projective geometry