Matrices: new axes, not a table of numbers

Most books introduce a matrix as a table of numbers with a “row times column” rule. This page starts from the other side: a matrix is a set of new axes, written as columns. From there you can see what each number does, what the determinant means, why order matters and how to get back to where you started.

Why this matters

A matrix places every object in a scene: it turns a rock, stretches a tree and holds a lantern in a hand. The graphics card carries each vertex into the world and onto the screen by multiplying it by a matrix. The lesson on spaces introduced the matrix as “axes and a pivot.” This page covers the math behind it: where the rules come from and what follows from them. This page builds on the lessons on vectors and on sine and cosine.

Further in the article

1. A matrix is where the axes went
2. Three basic matrices
3. The determinant: how much the area changes
4. Matrix products: do one thing, then the other
5. Translation does not fit in 2 × 2
6. The inverse matrix: getting back
7. Hierarchy: a chain of products
8. Where it breaks
9. Connections: the same idea in other fields
10. Key points
11. Test yourself