A 2×2 matrix morphs the plane through stretch, reflection, shear, and rotation.
From the chapter: Chapter 2: Linear Algebra
Glossary: matrix, linear map, determinant
Transcript
A matrix is not a table of numbers. It is a recipe for moving the plane.
This is the identity. Every point stays exactly where it is, and the basis vectors î and ĵ point along their usual axes.
Now the top-left entry grows. Space stretches horizontally while the vertical direction is left untouched.
Next, the bottom-right entry flips negative. The plane reflects top to bottom.
An off-diagonal entry appears. Vertical lines tilt, and the grid skews. This is a shear.
Finally, all four entries take the values cosine and sine of thirty degrees. The whole plane spins. This is a rotation.
Every familiar transformation, rotation, shear, reflection, scale, is one specific choice of those four numbers.