Labs / Mathematics
Matrix Transformations
A matrix is just an instruction for moving space. Drag the sliders for the four entries a, b, c & d and watch the grid, the basis vectors î and ĵ, and the F shape bend along with them. Grab the F to slide it around, and read off the determinant — the factor by which every area is scaled.
M [1.00 1.00; 0.00 1.00]det 1.00invertible yes
What to try
- Where do î and ĵ end up? Notice that they are always exactly the two columns of the matrix.
- Hit Shear and check the determinant. Why does the F look slanted yet still cover the same area?
- Drive the determinant to zero. What happens to the whole grid, and why does “invertible” flip to no?
- Make the determinant negative with Reflect. How can you tell the plane has been flipped over rather than just turned?
- Combine a rotation with a stretch by hand — can you build a transformation that turns the F upside down?