5 Lines
TOOD: Figure out how to write blueprints about definitions
A line over a field \(\mathbb {F}\) is a linear subspace of \(\mathbb {F}^3\) of dimension 2.
A point \((x, y) \in \mathbb {F}^2\) is in a line \(L\) iff \((x, y, 1) \in L\).
Given a linear isomorphism \(P\) and a line \(L\), we have a line \(PL\).
Proof
This is a valid line because linear isomorphism preserves dimension.
For any linear equivalence, applying it to lines is injective.
Proof
From the injectivity of linear isomorphisms.
Given a set \(P\) of points and a set \(L\) of lines, the number of incidences is at most \(\sqrt{|L| |P| (|P| + |L|)}\).
TODO2