Change of Coordinates and Projective Linear Group
Affine Change of Coordinates
In affine space $\mathbb{A}^n(k)$, a change of coordinates is given by an affine linear transformation:
\[\vec{x} \mapsto A \vec{x} + \vec{b}\]where $A \in \mathrm{GL}_n(k)$ is an invertible $n \times n$ matrix and $\vec{b} \in k^n$ is a translation vector. Such transformations form the affine group:
\[\mathrm{Aff}_n(k) = \mathrm{GL}_n(k) \ltimes k^n\]This group acts on $\mathbb{A}^n(k)$ and preserves affine properties like collinearity and ratios of distances along lines.
Projective Change of Coordinates
In projective space $\mathbb{P}^n(k)$, a change of coordinates corresponds to applying an invertible linear transformation of $k^{n+1}$:
\[(x_0 : x_1 : \cdots : x_n) \mapsto (x_0', x_1', \ldots, x_n') = A(x_0, x_1, \ldots, x_n)^T\]where $A \in \mathrm{GL}_{n+1}(k)$. However, since projective points are only defined up to scalar, two matrices $A$ and $\lambda A$ (for $\lambda \in k^\times$) induce the same projective transformation. Thus, the group of projective coordinate changes is:
\[\mathrm{PGL}_{n+1}(k) = \mathrm{GL}_{n+1}(k) / k^\times\]This is called the projective general linear group.
Action on Projective Space
An element of $\mathrm{PGL}_{n+1}(k)$ acts on $\mathbb{P}^n(k)$ by:
\[A \cdot (x_0 : x_1 : \cdots : x_n) = (x_0' : x_1' : \cdots : x_n')\]where $(x_0’, \ldots, x_n’) = A (x_0, \ldots, x_n)^T$, and the result is interpreted modulo scalar. This action:
- Preserves incidence relations (e.g., whether points lie on a line),
- Maps lines to lines,
- Is transitive on triples of points in general position (when $n = 1$).
Example: Projective Line $\mathbb{P}^1(k)$
When $n = 1$, $\mathrm{PGL}_2(\mathbb{C})$ acts on $\mathbb{P}^1(\mathbb{C}) = \mathbb{C}\cup\{\infty\}$ by fractional linear transformations:
\[\begin{pmatrix} a & b \\ c & d \end{pmatrix} \cdot (x : y) = (ax + by : cx + dy)\]In affine coordinates $z = x/y$, this becomes the familiar Möbius transformation:
\[z \mapsto \frac{az + b}{cz + d}\]with $ad - bc \ne 0$.