4.10 Linear Transformations and Matrices

Matrices as functions

A 2x2 matrix can act as a function that transforms points or vectors in the plane. If matrix A maps vector v to vector w, we write Av = w. This transformation sends the plane to a new plane, changing shapes by rotating, reflecting, stretching, shrinking, or combinations of these.

Effect on unit square & basic transformations

A matrix's columns show where the basis vectors (1,0) and (0,1) land after transformation. This lets you predict how any shape, like the unit square, is stretched, rotated, or reflected. Common transformations include rotations, reflections, dilations, and shears, each with a characteristic matrix form.

Determinant and area change

The determinant of a transformation matrix tells you how area scales under the transformation. If det(A) = k, areas are multiplied by |k|. A negative determinant means the transformation also flips orientation (like a reflection). A determinant of 0 means the transformation collapses area to a line or point.

Key terms

Linear transformation

A function represented by a matrix that maps vectors from one space to another, preserving vector addition and scalar multiplication.

Basis vectors under transformation

The columns of a transformation matrix show where (1,0) and (0,1) are mapped, defining the entire transformation.

Determinant and area

The absolute value of the determinant of a matrix gives the factor by which areas are scaled under the transformation.

Negative determinant

Indicates the transformation reverses orientation (a reflection), in addition to any scaling.

Determinant of 0

Means the transformation collapses the plane onto a line or point, and the matrix is not invertible.

Try an AP-style question

What does [[0, −1], [1, 0]] do?

1. Reflects across the x-axis
2. Rotates vectors 90° counterclockwise
3. Rotates vectors 90° clockwise
4. Scales by −1