Matrices as Machines

Transforming the whole plane

An interactive lesson with 7 steps, about 6 minutes: you predict, drag and solve, with instant feedback on every answer. It runs in your browser and needs JavaScript.

What you’ll learn

• A matrix’s columns show where the basis vectors land. Grid lines stay straight and parallel.
• $\begin{pmatrix} a & b \\ c & d \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} ax+by \\ cx+dy \end{pmatrix}$.
• The determinant $ad - bc$ is the area scale factor. If it's 0, the plane is squashed flat.

Steps

1. Where do the arrows go?
2. Multiplying a vector
3. Transform a vector
4. Rotation
5. The whole grid moves
6. The determinant
7. Area scale