The group $PGL(n,\mathbb R)$ is the quotient group $GL(n,\mathbb R)/Z$ where $Z$ is the set of $n\times n$ scalar matrices.
This Wikipedia article says that $PGL$ takes collinear points to collinear points. It also says that $PGL$ acts on faithfully on projective space. These two statements seem contradictory to me. A point in projective space is actually a line and hence should be mapped to itself via the action of any element from $PGL$. Doesn't that contradict faithfulness? What am I missing?
Also, how does one actually prove that "$PGL$ takes collinear points to collinear points"?
Thank you.
Perhaps the confusion comes from the fact that the notion of "point" has not been clarified. I think what is meant here by a "point" is a point in projective space $\Bbb RP^{n-1}$, i.e., a 1-dimensional linear subspace of $\Bbb R^n$. With that understanding, the comment becomes clear: an element of $PGL(n)$ is reprented by a matrix which acts linearly on $\Bbb R^n$. Therefore if a triple of points is collinear, i.e., the corresonding 1-dimensional subspaces of $\Bbb R^n$ lie in a common 2-dimensional subspace of $\Bbb R^n$, then their images will also lie in a 2-dimensional subspace, i.e., a common projective line.