AP PRECALCULUS — UNIT 4B · FUNCTIONS INVOLVING PARAMETERS, VECTORS, AND MATRICES
4.13A — Matrices as Functions
Notes — Inputs, Outputs, and Compositions
💡 Learning Objectives (4.13.A Part 1)
By the end of this lesson you will be able to:
- Treat a 2 × 2 matrix as a function from ℝ² to ℝ²
- Compute the image of a vector under a matrix function
- Compose matrix functions and recognize the composition as a product
- Reverse a matrix function using the inverse matrix
1. The Function Viewpoint
In Topic 4.12 we saw matrices as transformations. We can be a bit more formal: every 2 × 2 matrix A defines a FUNCTION:
T_A: ℝ² → ℝ², T_A(v) = Av
INPUT: a 2D vector v. OUTPUT: another 2D vector Av. Like any function, we can compose, invert, and study key features.
2. Image and Pre-Image
The IMAGE of v under T_A is just T_A(v) = Av. The PRE-IMAGE of a vector w under T_A is the set of vectors v with T_A(v) = w, i.e., the solution set of Av = w.
📘 Example — Pre-image
A = [[2, 1], [0, 3]], find the pre-image of [[5], [6]].
- Solve Av = [[5], [6]]: 2x + y = 5, 3y = 6 ⇒ y = 2, x = 3/2
- Pre-image: v = [[3/2], [2]]
- Check: A · [[3/2], [2]] = [[3 + 2], [0 + 6]] = [[5], [6]] ✓
3. Composition of Matrix Functions
If T_A and T_B are matrix functions, their COMPOSITION (T_B ∘ T_A)(v) = T_B(T_A(v)) is itself a matrix function:
(T_B ∘ T_A)(v) = B(Av) = (BA) v
So composition CORRESPONDS to MATRIX MULTIPLICATION. The matrix of the composed function is the product BA — note the order: do A first, write B on the left.
📘 Example — Compose two transformations
T_A reflects over the x-axis: A = [[1, 0], [0, −1]].
T_B rotates 90° CCW: B = [[0, −1], [1, 0]].
- Compose ‘reflect, then rotate’: BA = [[0, −1], [1, 0]] · [[1, 0], [0, −1]] = [[0, 1], [1, 0]]
- This is the matrix for reflection over y = x — a single transformation that achieves the same as the two combined.
4. Identity as Function
The identity matrix I corresponds to the IDENTITY FUNCTION T_I(v) = v. It does nothing.
5. Inverse Functions
If A is invertible (det(A) ≠ 0), then T_A is one-to-one and onto, with inverse function T_(A⁻¹). That is:
T_(A⁻¹)(T_A(v)) = v and T_A(T_(A⁻¹)(v)) = v
So A⁻¹ undoes the transformation A.
6. When the Function Is Not One-to-One
If det(A) = 0, then T_A is NOT one-to-one — distinct inputs can produce the same output. The output set (range) is a line through the origin or just the origin itself.
📘 Example — Non-injective transformation
A = [[1, 1], [2, 2]] — det = 0.
- T_A([[3], [0]]) = [[3], [6]]
- T_A([[2], [1]]) = [[3], [6]]
- Different inputs, same output — not one-to-one. All outputs lie on the line y = 2x.
7. Function Notation in Practice
Sometimes matrix functions are written with a more familiar notation. If we write T(x, y) for the function that takes the input (x, y) and outputs T(x, y) as a column, then:
- T(x, y) = (ax + by, cx + dy) corresponds to A = [[a, b], [c, d]]
- ‘Apply T’ is the same as ‘multiply by A’
- Composing two such functions T₁ and T₂ corresponds to multiplying their matrices
8. Summary
- A matrix A defines a function T_A(v) = Av from ℝ² to ℝ²
- Pre-image of w: solve Av = w for v
- Composition of matrix functions corresponds to MATRIX MULTIPLICATION
- Identity function is given by the identity matrix
- Inverse function exists iff det(A) ≠ 0; it is given by A⁻¹
- If det(A) = 0, T_A is not one-to-one; output is at most a line