AP PRECALCULUS — UNIT 4B · FUNCTIONS INVOLVING PARAMETERS, VECTORS, AND MATRICES
4.12 — Linear Transformations and Matrices
Notes — Matrices that Move Points in the Plane
💡 Learning Objectives (4.12.A)
By the end of this lesson you will be able to:
- Apply a 2 × 2 matrix to a 2D vector to produce a transformed vector
- Identify the geometric effect of basic matrices: rotations, reflections, scaling, shears
- Compose transformations by multiplying matrices
- Connect the determinant to the area-scaling factor of a transformation
1. Matrix Times Vector
If A is a 2 × 2 matrix and v is a 2 × 1 column vector, the product Av is a new 2 × 1 column vector — a TRANSFORMED version of v. We think of the matrix A as a FUNCTION that takes vectors and outputs vectors.
📘 Example — Apply a transformation
A = [[2, 0], [0, 3]], v = [[5], [1]].
- Av = [[2·5 + 0·1], [0·5 + 3·1]] = [[10], [3]]
- Geometrically: x stretched by 2, y stretched by 3 — a horizontal/vertical scaling.
2. Standard Transformations
💡 Common 2 × 2 Matrices
- Identity: [[1, 0], [0, 1]] — leaves every vector unchanged
- Scaling by k: [[k, 0], [0, k]] — uniform stretch by factor k
- Horizontal scale by a: [[a, 0], [0, 1]]
- Vertical scale by b: [[1, 0], [0, b]]
- Reflection over x-axis: [[1, 0], [0, −1]]
- Reflection over y-axis: [[−1, 0], [0, 1]]
- Reflection over y = x: [[0, 1], [1, 0]]
- Rotation by θ counter-clockwise: [[cos θ, −sin θ], [sin θ, cos θ]]
3. Worked Examples
📘 Example — Rotate a vector by 90°
Rotation matrix for 90° CCW: [[cos 90°, −sin 90°], [sin 90°, cos 90°]] = [[0, −1], [1, 0]].
- Apply to v = [[3], [4]]: [[0·3 − 1·4], [1·3 + 0·4]] = [[−4], [3]]
- Geometric check: rotating (3, 4) by 90° CCW should give (−4, 3) ✓
📘 Example — Reflect over y-axis
Matrix: [[−1, 0], [0, 1]]. Apply to v = [[2], [5]]:
- [[−2], [5]] — x flipped, y unchanged ✓
4. Composing Transformations
Applying transformation A and then transformation B is the same as applying the matrix BA (note the order). This is because B(Av) = (BA)v. ORDER MATTERS — do A first, then B, but write BA in the formula.
📘 Example — Rotate then scale
Apply 90° rotation R = [[0, −1], [1, 0]], then scale by 2 with S = [[2, 0], [0, 2]].
- Combined matrix: SR = [[2, 0], [0, 2]] · [[0, −1], [1, 0]] = [[0, −2], [2, 0]]
- This rotates 90° CCW AND scales by 2.
5. The Determinant as Area Scaling
When you apply A to the unit square (with corners at (0,0), (1,0), (0,1), (1,1)), you get a parallelogram. Its area is exactly |det(A)|. So:
- |det(A)| > 1: transformation EXPANDS area
- |det(A)| < 1: transformation CONTRACTS area
- |det(A)| = 1: transformation PRESERVES area (e.g. rotations and reflections)
- det(A) > 0: orientation preserved (counter-clockwise stays CCW)
- det(A) < 0: orientation reversed (CCW becomes CW)
- det(A) = 0: transformation collapses everything to a line or point — not invertible
📘 Example — Area scaling
A = [[2, 0], [0, 3]]: det = 6, so areas multiply by 6.
B = [[1, 0], [0, −1]] (reflection): det = −1, areas preserved but orientation flipped.
C = [[2, 4], [1, 2]]: det = 4 − 4 = 0, transformation collapses the plane to a line.
6. Inverse Transformations
If A represents a transformation and is invertible, then A⁻¹ represents the REVERSE transformation: applying A then A⁻¹ returns every vector to where it started.
- A rotation by θ has inverse: rotation by −θ
- A scaling by k has inverse: scaling by 1/k
- A reflection has inverse: itself (reflecting twice returns to start)
7. What Linear Transformations Preserve
Every transformation defined by a matrix multiplication has these properties:
- The origin maps to the origin (since A · 0 = 0)
- Lines map to lines (or points, if det = 0)
- Parallel lines stay parallel
- Ratios of lengths along the same line are preserved
- Linear combinations are preserved: A(u + v) = Au + Av and A(kv) = k(Av)
These properties are what make these called LINEAR transformations.
8. Summary
- A 2 × 2 matrix acts as a function on vectors: v ↦ Av
- Standard matrices encode rotations, reflections, scalings, and shears
- Compose transformations by multiplying matrices in REVERSE order: B then A is AB
- |det(A)| is the area-scaling factor; sign tells you about orientation
- Linear transformations always send the origin to the origin and lines to lines