Linear Transformations and Matrices

Unit 4 · Functions Involving Parameters, Vectors, and Matrices

What AP Precalc asks here

A linear transformation $T: \mathbb{R}^2 \to \mathbb{R}^2$ is captured by a 2×2 matrix: $T(\vec v) = A \vec v$ . Common transformations have recognizable matrices: rotations involve $\sin\theta, \cos\theta$ ; reflections have $\pm 1$ on the diagonals; scalings have $k$ on the diagonal. Composing two transformations corresponds to multiplying their matrices.

Standard 2D transformations

Scaling by k

\begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix}

Reflection over x-axis

\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}

Rotation by

\theta

\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}

AP Tip: Composition order matters:

T_2(T_1(\vec v)) = A_2 (A_1 \vec v) = (A_2 A_1) \vec v

. The matrix applied first goes on the right.

AP problem types

Type 1

Apply T(v) = Av

Compute the matrix-vector product as in Topic 4.10.

Example 1

T

corresponds to

\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}

, find

T((1, 0))

. (A)

(0, 1)

(B)

(1, 0)

(C)

(-1, 0)

(D)

(0, -1)

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Type 2

Recognize common transformations

Match the matrix form to a named transformation.

Example 2

Which transformation does

\begin{pmatrix} 3 & 0 \\ 0 & 3 \end{pmatrix}

represent? (A) Rotation by 90° (B) Reflection across y-axis (C) Scaling by factor 3 (D) Shear

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Type 3

Compose transformations

Multiply the two matrices in the right order, then apply to the vector.

Example 3

Apply rotation by 90° (matrix

R

) then reflection across x-axis (matrix

F

) to

(1, 0)

. What is the result? (A)

(1, 0)

(B)

(0, 1)

(C)

(0, -1)

(D)

(-1, 0)

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