18 min read

Introduction to Matrix Transformations

Multiplying a coordinate point by a special matrix moves it geometrically — reflecting, rotating, or scaling it.

Moving Points With Matrices

Transform lab

Scale factor 1.0Rotation 0° (anticlockwise)

P = (3, 2) · scaled ≈ (3.00, 2.00) · rotated ≈ (3.00, 2.00) · scale then rotate ≈ (3.00, 2.00)

Change a value and watch:

Adjust the scale and angle and watch the point move. This is exactly what a transformation matrix does, written as a single calculation.

Every transformation you will meet in this section can be written as a 2 \times 2 matrix multiplying the point.

Example 1 — a coordinate-swap transformation

Apply the matrix (0110) to the point (3, 5). 1 Write the point as a column: (35) 2 Multiply: top entry = (0 \times 3) + (1 \times 5) = 5 . Bottom entry = (1 \times 3) + (0 \times 5) = 3 Answer: (5, 3) — this matrix reflects a point in the line y = x

Practice

What does the identity matrix (1001) do when applied to any point?