Matrices as Functions

Unit 4 · Functions Involving Parameters, Vectors, and Matrices

What AP Precalc asks here

Every $m \times n$ matrix $A$ defines a function $T: \mathbb{R}^n \to \mathbb{R}^m$ via $T(\vec v) = A \vec v$ . Standard function language carries over: domain ( $\mathbb{R}^n$ ), codomain ( $\mathbb{R}^m$ ), image (the set of all $A \vec v$ ), and inverse (if $A$ is square and invertible, then $A^{-1}$ is the matrix of the inverse function).

Matrix function language

Function

T(\vec v) = A \vec v

Domain → Codomain

T: \mathbb{R}^n \to \mathbb{R}^m \text{ for } A \text{ of size } m \times n

Invertibility (square A)

A^{-1} \text{ exists} \iff \det A \neq 0

AP Tip:

|\det A|

equals the area scaling factor when

A

is 2×2. A unit square becomes a parallelogram of area

|\det A|

Type 1

Domain and codomain

An $m \times n$ matrix maps $\mathbb{R}^n$ to $\mathbb{R}^m$ .

Example 1

A 2×3 matrix A defines T(v) = Av. What is the domain and codomain? (A) Domain

\mathbb{R}^2

, codomain

\mathbb{R}^3

(B) Domain

\mathbb{R}^3

, codomain

\mathbb{R}^2

\mathbb{R}^2

, codomain

\mathbb{R}^2

(D) Domain

\mathbb{R}^3

, codomain

\mathbb{R}^3

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Type 2

Image of a set under a matrix

Apply A to each vector in the set.

Example 2

For

A = \begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix}

, what is the image of the unit square (vertices

(0,0), (1,0), (0,1), (1,1)

)? (A) The same unit square (B) A

2 \times 3

rectangle with vertices

(0,0), (2,0), (0,3), (2,3)

\sqrt{13}

(D) A triangle

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Type 3

Invertibility as inverse function

Compute the determinant. Nonzero ⇒ A has an inverse function $A^{-1}$ .

Example 3

Is the function

T(v) = \begin{pmatrix} 2 & 4 \\ 1 & 2 \end{pmatrix} v

invertible as a function? (A) Yes (B) No, because det = 0 (rows are proportional) (C) No, because the codomain has a different dimension (D) Cannot be determined

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