Matrix Transform Lab
Skill goal: See what a 2×2 matrix does to the plane: rotations, reflections, shears, and squashes — and what the determinant means.
1 · Observe
Light grid: the original plane. Blue grid: after the matrix acts. Watch the unit square.
2 · Manipulate
3 · Predict
Set the “Collapse” preset. Before looking: what happens to the whole grid? What is the determinant, and what does that tell you about area?
4 · See
5 · Explain
A matrix transforms every vector v into Mv — straight lines stay straight, and the origin stays put. The determinant is the area scale factor: det = 2 doubles every area, det = −1 flips orientation (reflection), det = 0 squashes the plane flat (the matrix is not invertible). Columns of M are where the basis vectors î and ĵ land.
6 · Challenge
Q: A matrix with det = 0…