Family Overview

In Plain Language

Topic 4.13 treats matrices as functions that transform vectors in R^2.

• Association: each linear transformation corresponds to a 2x2 matrix (4.13.A.1).
• Basis images: where (1,0) and (0,1) go determines the matrix (4.13.A.2).
• Rotation: [[cos t, -sin t],[sin t, cos t]] rotates counterclockwise by t (4.13.A.3).
• Dilation: |det(A)| gives the magnitude of area scaling under A (4.13.A.4).
• Composition: composing transformations is linear; matrix is the product (4.13.B.1, 4.13.B.2).
• Inverse: inverse transformations undo each other; L^-1(v)=A^-1 v when L(v)=Av (4.13.C.1, 4.13.C.2).

Course scope only (Unit 4 additional topic; not on the AP Exam).

Why It Matters

Matrices-as-functions unify geometry (rotate, scale, shear) with algebra and prepare for 4.14 modeling.

How Families Can Help

• Ask where the unit vectors land.
• Compare before/after areas when |det| is discussed.
• Check whether order matters in composition.

What Not to Do

• Do not swap rotation matrix signs casually.
• Do not assume AB=BA for transformations.
• Do not skip linking matrix entries to basis images.