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.

The fastest way to understand a machine is to feed it something simple and watch. Our test object is the unit square, the square with corners $(0,0)$, $(1,0)$, $(0,1)$, $(1,1)$. Its image under $A$ turns out to be a parallelogram whose edges are precisely the columns of $A$ — so a matrix is not a table of four unrelated numbers but a pair of instructions: where the first basis vector goes, and where the second one goes. Once you read matrices column-by-column, you can predict the fate of any grid, any square, any shape.

In this chapter you will drive the four entries of a matrix with sliders and watch the unit square shear, stretch, and flatten in response; then you will take a single matrix and morph the identity into it, watching the plane deform continuously along the way. From there the questions sharpen: which outputs can a transformation actually reach, and can two different inputs ever collide? What single matrix describes doing one transformation and then another — and does the order matter? And when can the motion be run backwards? Riding along in every one of these figures is a quantity that silently keeps score of how much area survives the trip — Chapter 4 gives it a name.