Rotation Matrices

A matrix spins the whole plane because it's defined by where it sends just two points: (1, 0) and (0, 1). If you decide those basis vectors land at (cos theta, sin theta) and (-sin theta, cos theta), every other point is forced to rotate by theta too, since any point (p, q) is just p times the first basis vector plus q times the second. That's the whole trick: nail down the two unit vectors, and linearity does the rest.

This gives the rotation matrix [[cos theta, -sin theta], [sin theta, cos theta]]. Multiply it by (p, q) and you get x' = p cos theta - q sin theta, y' = p sin theta + q cos theta. The columns of the matrix are literally the images of (1,0) and (0,1), which is why building rotation matrices is just a matter of knowing those two landing spots.

Rotation matrices sit next to the other matrix transformations in this unit: scaling, reflecting, shearing. What makes rotation special is that it preserves shape and size perfectly, it only changes orientation in the plane. That shows up in the determinant: for any rotation matrix, cos theta times cos theta minus (-sin theta) times sin theta equals cos² theta + sin² theta = 1. A determinant of 1 means areas are preserved exactly and orientation is kept, unlike a reflection or a scaling matrix whose determinant stretches or flips area.