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 go to straight lines or collapse to single points, parallel lines stay parallel wherever they survive, 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.