Reading through my notes of Affine Geometry I stumbled upon an exercise which cites the concept of a homology and I wish to understand how -in the context of the exercise- a homology is defined. The exercise reads:
"Let $f$ be a general homology of the $s$ axis, direction $u$ and ratio $r$, and let $g$ be a general homology of the affine plane of the $t$ axis, parallel to $s$, of direction $v$ and ratio $R$. (Remember that a homology is a bijective affine map from the affine plane with a line of fixed points. The ratio $\rho$ of a homology is its determinant. If $\rho = 1$ we say that the homology is special, otherwise we say it is general.) Let $h = g\circ f$ and show that:
$h$ is a general homology if and only if $Rr\neq 1$ and $\langle u \rangle = \langle v \rangle$.
$h$ is a translation if and only if $Rr=1$ and $\langle u \rangle = \langle v \rangle$."
I find the statment a bit vague and wanted to clear some things up.
Is the $s$ axis supposed to be a line of the form $s_0+\langle u\rangle$?
Does the homology being a map "from a plane" mean that the domain of the homology is a plane?
Which space are we working on? Is, for example, its dimension implicit in the statement or is the exercise expecting us to consider a space of arbitrary dimensions?