Language used in projective linear group In lectures and text on topic of projective linear group, I hear and see the word "factor out" or "quotient out" thrown around a lot. What is the word supposed to mean?
If this is vague, I can provide examples. 
 A: Generally, if $X$ is a set and $R$ is an equivalence relation on $X$, the set $X/R$ of equivalence classes is the quotient of $X$ by $R$, and is said to be obtained by factoring out $R$. The mapping $\Pi:X \to X/R$ defined by $\Pi(x) = [x]$ is called the quotient map or projection associated to the relation $R$.
In this setting, a mapping $f:X \to Y$ factors through the quotient if $f$ is constant on $R$-equivalence classes, i.e., if for all $x$ and $x'$ in $X$, $xRx'$ implies $f(x) = f(x')$. In this event, there exists a unique mapping $\bar{f}:X/R \to Y$, defined by $\bar{f}([x]) = f(x)$, that satisfies $f = \bar{f} \circ \Pi$. Factoring here refers to writing the mapping $f$ as a composition.
The $n$-dimensional projective space over a field $K$, for example, is obtained from the set $X = K^{n+1} \setminus\{0\}$ of non-zero ordered $(n + 1)$-tuples from $K$ by factoring out by the action of $K^{\times}$, the multiplicative group of non-zero elements of $K$. That is, two $(n + 1)$-tuples $x$ and $x'$ in $X$ are equivalent if there exists a non-zero element $k$ in $K$ such that $x' = kx$. The set of equivalence classes is precisely the set of one-dimensional subspaces in $K^{n+1}$, a.k.a., "the set of lines through the origin".
Ratios of homogeneous coordinates factor through the quotient, and therefore define "affine coordinates" on projective space, though the homogeneous coordinates themselves (Cartesian coordinates on $K^{n+1}$) do not.
Similarly, in the general linear group $GL(n + 1, K)$, you can view two transformations as equivalent if one is a scalar multiple of the other. As Tobias Kildetoft comments, here you're considering the set $K^{\times} I$ of scalar transformations, which constitute a normal subgroup of $GL(n + 1, K)$, and forming the set of cosets $GL(n + 1, K)/K^{\times}I$ to get the projective linear group $PGL(n + 1, K)$.
