What is the image of bmatrix 1 \\ 0 bmatrix under the matrix bmatrix 2 & 5 \\ 3 & 4 bmatrix? [1 point]

College Board AP 2025 (style)-style practice: Section II (free response, no calculator). (a) Write the matrix that reflects vectors over the x-axis, and apply it to bmatrix 2 \\ 5 bmatrix. (b) Write the matrix that rotates vectors 90^ counterclockwise, and apply it to bmatrix 1 \\ 0 bmatrix.

A 4-point question on standard transformation matrices. (a) Reflection (2 points): reflecting over the x-axis negates the y-component, bmatrix 1 & 0 \\ 0 & -1 bmatrix. Applied: bmatrix 1 & 0 \\ 0 & -1 bmatrixbmatrix 2 \\ 5 bmatrix = bmatrix 2 \\ -5 bmatrix. (b) Rotation (2 points): a 90^ counterclockwise rotation is bmatrix 0 & -1 \\ 1 & 0 bmatrix. Applied: bmatrix 0 & -1 \\ 1 & 0 bmatrixbmatrix 1 \\ 0 bmatrix = bmatrix 0 \\ 1 bmatrix, sending the right-pointing vector to the up-pointing one.

United StatesPrecalculusSyllabus dot point

How does a matrix represent a linear transformation of the plane, such as a rotation, reflection or scaling?

Topic 4.12 Linear Transformations and Matrices: represent a linear transformation of the plane by a matrix, and identify the matrices for scalings, reflections and rotations.

A focused answer to AP Precalculus Topic 4.12, covering how a 2x2 matrix represents a linear transformation, how the columns are the images of the basis vectors, and the standard matrices for scalings, reflections and rotations.

Generated by Claude Opus 4.89 min answerUpdated 2026-06-04

Reviewed by: AI editorial process; not yet individually human-reviewed

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Quick answer

A $2 \times 2$ matrix acts on the plane as a linear transformation: multiply any vector by the matrix to get its image. The transformation is completely determined by where it sends the two basis vectors $\begin{bmatrix} 1 \\ 0 \end{bmatrix}$ and $\begin{bmatrix} 0 \\ 1 \end{bmatrix}$ , and those images are exactly the first and second columns of the matrix. So you can build a transformation matrix by writing down where the basis vectors should go. The standard transformations have memorable matrices: uniform scaling by $k$ is $\begin{bmatrix} k & 0 \\ 0 & k \end{bmatrix}$ ; reflection over the $x$ -axis is $\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$ ; rotation by angle $\theta$ counterclockwise is $\begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}$ . Composing transformations corresponds to multiplying their matrices, in the order the transformations are applied.

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What this topic is asking
A matrix as a transformation
Columns are the images of the basis vectors
The standard transformation matrices
Composing transformations
Try this

What this topic is asking

The College Board (Topic 4.12) wants you to see a $2 \times 2$ matrix as a linear transformation of the plane: a rule that sends each vector to a new vector by matrix multiplication. You should know that the matrix's columns are the images of the basis vectors, and recognize the standard matrices for scalings, reflections and rotations.

A matrix as a transformation

So "transform the plane" and "multiply by this matrix" are the same operation; the matrix is the transformation written as numbers.

Columns are the images of the basis vectors

This is the quickest way to construct a transformation matrix: decide the fate of the two basis vectors and read off the columns.

The standard transformation matrices

Building a rotation and applying it

Find the matrix for a $90^\circ$ counterclockwise rotation, and use it to rotate $\begin{bmatrix} 3 \\ 1 \end{bmatrix}$ .

step 1 Substitute the angle into the rotation matrix

With $\theta = 90^\circ$ , $\cos 90^\circ = 0$ and $\sin 90^\circ = 1$ , so the matrix is $\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$ .

step 2 Confirm via the basis vectors

The first column $\begin{bmatrix} 0 \\ 1 \end{bmatrix}$ is the image of $\begin{bmatrix} 1 \\ 0 \end{bmatrix}$ (right turns to up); the second column $\begin{bmatrix} -1 \\ 0 \end{bmatrix}$ is the image of $\begin{bmatrix} 0 \\ 1 \end{bmatrix}$ (up turns to left). Both are $90^\circ$ counterclockwise.

step 3 Apply the matrix to the vector

$\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}\begin{bmatrix} 3 \\ 1 \end{bmatrix} = \begin{bmatrix} 0\cdot3 + (-1)\cdot1 \\ 1\cdot3 + 0\cdot1 \end{bmatrix} = \begin{bmatrix} -1 \\ 3 \end{bmatrix}$ .

step 4 Interpret

The vector $\begin{bmatrix} 3 \\ 1 \end{bmatrix}$ rotates to $\begin{bmatrix} -1 \\ 3 \end{bmatrix}$ , the same length but turned a quarter turn counterclockwise.

Composing transformations

A point worth stating once is that performing one transformation after another corresponds to multiplying their matrices, with the first transformation on the right. If you rotate then reflect, the combined matrix is (reflection)(rotation), because the rightmost matrix acts on the vector first. This is exactly why matrix multiplication is not commutative (Topic 4.10): reversing the order of two transformations generally gives a different result, just as putting on socks then shoes differs from shoes then socks. Reading a product of matrices right-to-left as a sequence of transformations connects the algebra of Topic 4.10 to the geometry here and prepares the function view of Topic 4.13.

Try this

Q1. What matrix reflects vectors over the $y$ -axis? [1 point]

Cue. $\begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}$ , negating the $x$ -component.

Q2. What is the image of $\begin{bmatrix} 1 \\ 0 \end{bmatrix}$ under the matrix $\begin{bmatrix} 2 & 5 \\ 3 & 4 \end{bmatrix}$ ? [1 point]

Cue. The first column, $\begin{bmatrix} 2 \\ 3 \end{bmatrix}$ .

Exam-style practice questions

Practice questions written in the style of College Board exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

AP 2025 (style)1 marksSection I, Part A (multiple choice, no calculator). Which matrix scales every vector in the plane by a factor of

3

? (A)

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

(B)

\begin{bmatrix} 3 & 3 \\ 3 & 3 \end{bmatrix}

(C)

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

(D)

\begin{bmatrix} 1 & 0 \\ 0 & 3 \end{bmatrix}

Show worked answer →

The correct answer is (A), $\begin{bmatrix} 3 & 0 \\ 0 & 3 \end{bmatrix}$ .

A uniform scaling by $k$ is $kI = \begin{bmatrix} k & 0 \\ 0 & k \end{bmatrix}$ . Multiplying any vector by it gives $\begin{bmatrix} 3x \\ 3y \end{bmatrix}$ , scaling both components by $3$ . Choice (D) scales only the $y$ -component.

AP 2025 (style)4 marksSection II (free response, no calculator). (a) Write the matrix that reflects vectors over the

x

-axis, and apply it to

\begin{bmatrix} 2 \\ 5 \end{bmatrix}

. (b) Write the matrix that rotates vectors

90^\circ

counterclockwise, and apply it to

\begin{bmatrix} 1 \\ 0 \end{bmatrix}

Show worked answer →

A 4-point question on standard transformation matrices.

(a) Reflection (2 points): reflecting over the $x$ -axis negates the $y$ -component, $\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$ . Applied: $\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}\begin{bmatrix} 2 \\ 5 \end{bmatrix} = \begin{bmatrix} 2 \\ -5 \end{bmatrix}$ .
(b) Rotation (2 points): a $90^\circ$ counterclockwise rotation is $\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$ . Applied: $\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}\begin{bmatrix} 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 1 \end{bmatrix}$ , sending the right-pointing vector to the up-pointing one.

Related dot points

Sources & how we know this

AP Precalculus Course and Exam Description — College Board (2023)