Chapter 3: Linear Transformations: A Matrix Moves the Plane

So far a matrix has been bookkeeping — a grid of coefficients read off a system of equations. This chapter promotes it to a verb. A $2\times 2$ matrix $A$ acts: it takes each point $(x,y)$ of the plane and moves it to a new point, and it does so with a discipline no arbitrary motion has. The origin stays fixed, straight lines stay straight, parallel lines stay parallel, and evenly spaced points stay evenly spaced. Every such motion — rotation, stretch, shear, reflection, collapse — is a linear transformation, and every linear transformation of the plane is multiplication by some matrix.