Part IV · Planes from fields · Lesson 10

Vectors over a finite field

GF(q)³ as a finite grid of vectors, and counting the lines and planes through the origin.

This lesson is planned. Here is what it will cover.

You’ll learn

• Vectors $(x, y, z)$ with entries in $\mathrm{GF}(q)$: $q^3$ of them, a finite 3D grid.
• Spans, subspaces, and why every line through the origin holds $q - 1$ nonzero vectors.
• Counting: $\frac{q^3 - 1}{q - 1} = q^2 + q + 1$ lines through the origin, and the same number of planes.