Unit 9
Vectors and transformations.
A matrix as a map of the plane: turns, stretches, shears and flips, the determinant, and symmetry groups.
Before this unit
The unit, lesson by lesson
Lessons
3 lessons · 20 practice questions · about an hour
Matrices that move the plane
A linear map moves the whole plane at once, keeping lines straight and parallel: where the unit vectors go, how the rotation matrix is built, where any point goes, and why a shear does not change area.
The determinant
Every linear map scales every area by the same factor, the determinant ad − bc: why turns and shears keep area, what a negative determinant says, and what happens when it is zero.
Symmetry
A symmetry is a move that leaves a shape as it was: turns and reflections, how many symmetries a pattern has, how two reflections make a turn, and why a floor can repeat only turns of order 1, 2, 3, 4 or 6.