AP Precalculus Quiz: Linear Transformations And Matrices

What this quiz covers

This quiz focuses on Linear Transformations And Matrices, giving you a quick way to practice the rules, question types, and explanations that matter most for AP Precalculus.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. A function that maps an input vector to an output vector such that each component of the output vector is the sum of constant multiples of the input vector components (correct answer)
2. A function that maps an input vector to an output vector by adding a constant vector to the input vector components
3. A function that maps an input vector to an output vector by multiplying each component by the same scalar value
4. A function that maps an input vector to an output vector by rotating the input vector around the origin by a fixed angle

Explanation: A linear transformation is defined as a function that maps input vectors to output vectors such that each component of the output vector is the sum of constant multiples of the input vector components. This captures the essence of matrix multiplication where the transformation matrix determines these constant multiples.

Question 2

1. The result depends on the specific transformation matrix being used in the linear transformation
2. The result is always the zero vector regardless of the transformation matrix used (correct answer)
3. The result is always the unit vector in the direction of the first column of the matrix
4. The result is undefined because division by zero occurs in the transformation process

Explanation: A fundamental property of linear transformations is that they always map the zero vector to the zero vector. This is because when all input components are zero, any linear combination of these components will also be zero.

Question 3

1. As a $2 \times n$ matrix where each column represents one of the $n$ vectors in the set (correct answer)
2. As a $n \times 2$ matrix where each row represents one of the $n$ vectors in the set
3. As a single column vector containing all components of all vectors concatenated together in sequence
4. As a diagonal matrix with the vector components arranged along the main diagonal elements only

Explanation: A set of $n$ vectors in $\mathbb{R}^2$ is expressed as a $2 \times n$ matrix where each column represents one vector. This format allows a $2 \times 2$ transformation matrix to be multiplied with the $2 \times n$ matrix to transform all vectors simultaneously.

Question 4

1. There exists a unique $2 \times 2$ matrix $A$ such that $L(\vec{v}) = A\vec{v}$ for all vectors $\vec{v}$ in $\mathbb{R}^2$ (correct answer)
2. There exist multiple possible $2 \times 2$ matrices $A$ such that $L(\vec{v}) = A\vec{v}$ for vectors $\vec{v}$ in $\mathbb{R}^2$
3. The matrix $A$ must be square but can have dimensions other than $2 \times 2$ depending on the transformation
4. The relationship $L(\vec{v}) = A\vec{v}$ only holds for certain special vectors, not for all vectors in $\mathbb{R}^2$

Explanation: For any linear transformation $L$ from $\mathbb{R}^2$ to $\mathbb{R}^2$, there exists a unique $2 \times 2$ matrix $A$ such that $L(\vec{v}) = A\vec{v}$ for all vectors $\vec{v}$ in $\mathbb{R}^2$. This is a fundamental theorem about linear transformations and their matrix representations.

Question 5

1. A reflection across the $x$-axis that changes the sign of the $y$-coordinate while preserving the $x$-coordinate (correct answer)
2. A reflection across the $y$-axis that changes the sign of the $x$-coordinate while preserving the $y$-coordinate
3. A rotation by $90°$ counterclockwise that maps $(x,y)$ to $(-y,x)$ for all points in the plane
4. A scaling transformation that doubles the $x$-coordinate and halves the $y$-coordinate of every point

Explanation: The matrix $A = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$ represents a reflection across the $x$-axis. When applied to vector $\begin{bmatrix} x \\ y \end{bmatrix}$, it produces $\begin{bmatrix} x \\ -y \end{bmatrix}$, keeping the $x$-coordinate unchanged and negating the $y$-coordinate.

Question 6

1. The images of the unit vectors under the transformation become the columns of the transformation matrix (correct answer)
2. The images of the unit vectors under the transformation become the rows of the transformation matrix
3. The unit vectors determine the diagonal entries of the transformation matrix while other entries remain zero
4. The unit vectors determine the scaling factor that must be applied uniformly to all entries of the matrix

Explanation: The mapping of the unit vectors under a linear transformation provides the columns of the transformation matrix. If $L(\vec{e_1}) = \begin{bmatrix} a \\ c \end{bmatrix}$ and $L(\vec{e_2}) = \begin{bmatrix} b \\ d \end{bmatrix}$, then the transformation matrix is $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$.

Question 7

1. The magnitude of the dilation of regions in $\mathbb{R}^2$ under the transformation, indicating how areas change (correct answer)
2. The angle of rotation applied to all vectors in $\mathbb{R}^2$ under the transformation, measured in radians
3. The distance that all points in $\mathbb{R}^2$ are translated under the transformation in a fixed direction
4. The maximum scaling factor applied to any vector in $\mathbb{R}^2$ under the transformation along any direction

Explanation: The absolute value of the determinant of a $2 \times 2$ transformation matrix gives the magnitude of the dilation of regions in $\mathbb{R}^2$ under the transformation. It tells us by what factor areas are scaled when the transformation is applied.

Question 8

1. $A = \begin{bmatrix} 3 & -1 \\ 2 & 4 \end{bmatrix}$ (correct answer)
2. $A = \begin{bmatrix} 3 & 2 \\ -1 & 4 \end{bmatrix}$
3. $A = \begin{bmatrix} 1 & 0 \\ 3 & -1 \\ 2 & 4 \end{bmatrix}$
4. $A = \begin{bmatrix} 2 & 3 \\ 4 & -1 \end{bmatrix}$

Explanation: The transformation matrix is formed by placing the images of the unit vectors as columns. Since $\begin{bmatrix} 1 \\ 0 \end{bmatrix}$ maps to $\begin{bmatrix} 3 \\ 2 \end{bmatrix}$ and $\begin{bmatrix} 0 \\ 1 \end{bmatrix}$ maps to $\begin{bmatrix} -1 \\ 4 \end{bmatrix}$, the matrix is $A = \begin{bmatrix} 3 & -1 \\ 2 & 4 \end{bmatrix}$.

Question 9

1. The transformation changes areas by a factor of $6$ and reverses orientation of regions in the plane (correct answer)
2. The transformation rotates all vectors by $6$ radians counterclockwise about the origin without changing lengths
3. The transformation translates all points by $6$ units in the negative direction along both coordinate axes
4. The transformation scales all vectors by a factor of $-6$ uniformly in all directions from the origin

Explanation: A determinant of $-6$ means the transformation scales areas by a factor of $|{-6}| = 6$ and reverses orientation (because the determinant is negative). The negative sign indicates that the transformation flips the plane.

Question 10

1. $\begin{bmatrix} -4 \\ -3 \end{bmatrix}$, representing a reflection of the original vector across the $y$-axis (correct answer)
2. $\begin{bmatrix} 4 \\ 3 \end{bmatrix}$, representing a reflection of the original vector across the $x$-axis
3. $\begin{bmatrix} 3 \\ 4 \end{bmatrix}$, representing a $90°$ counterclockwise rotation of the original vector
4. $\begin{bmatrix} -3 \\ 4 \end{bmatrix}$, representing a $90°$ clockwise rotation of the original vector about the origin

Explanation: $\begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 4 \\ -3 \end{bmatrix} = \begin{bmatrix} -4 \\ -3 \end{bmatrix}$. This matrix reflects across the $y$-axis, negating the $x$-coordinate while keeping the $y$-coordinate unchanged.

Question 11

1. Linear transformations preserve vector addition and scalar multiplication: $L(\vec{u} + \vec{v}) = L(\vec{u}) + L(\vec{v})$ and $L(c\vec{v}) = cL(\vec{v})$ (correct answer)
2. Linear transformations always preserve the lengths of all vectors and the angles between any two vectors in the plane
3. Linear transformations can only rotate, reflect, or scale vectors but cannot perform any shearing or skewing operations
4. Linear transformations must have invertible matrices and therefore can always be reversed to recover the original vectors

Explanation: The defining properties of linear transformations are that they preserve vector addition and scalar multiplication. These two properties (linearity conditions) are what make a transformation linear, regardless of whether it preserves lengths, angles, or is invertible.

Question 12

1. A $2 \times n$ matrix containing the output vectors from applying transformation $A$ to each input vector (correct answer)
2. A $n \times 2$ matrix containing the output vectors arranged as rows rather than columns for easier interpretation
3. A $2 \times 2$ matrix representing the composition of the transformation with itself applied $n$ times successively
4. A $n \times n$ matrix showing all possible pairwise interactions between the input vectors under the transformation

Explanation: When a $2 \times 2$ transformation matrix $A$ multiplies a $2 \times n$ matrix of input vectors, the result is a $2 \times n$ matrix where each column is the transformed version of the corresponding input vector column.

Question 13

1. A scaling transformation that stretches vectors by factor $2$ in the $x$-direction and factor $3$ in the $y$-direction (correct answer)
2. A rotation transformation that rotates vectors by $2$ radians about the $x$-axis and $3$ radians about the $y$-axis
3. A translation transformation that moves all points $2$ units right and $3$ units up from their original positions
4. A shearing transformation that skews the plane by sliding points $2$ units horizontally and $3$ units vertically

Explanation: The diagonal matrix $A = \begin{bmatrix} 2 & 0 \\ 0 & 3 \end{bmatrix}$ represents a scaling transformation. It multiplies the $x$-coordinate by $2$ and the $y$-coordinate by $3$, stretching vectors differently in each direction.

Question 14

1. A rotation by angle $\theta$ counterclockwise about the origin that preserves distances and angles between vectors (correct answer)
2. A reflection across a line passing through the origin that makes an angle of $\theta$ with the positive $x$-axis
3. A scaling transformation that changes lengths by factor $\cos\theta$ in one direction and $\sin\theta$ in another direction
4. A shearing transformation that skews the plane by sliding points parallel to a line at angle $\theta$ to the $x$-axis

Explanation: The matrix $\begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}$ represents a rotation by angle $\theta$ counterclockwise about the origin. This is the standard rotation matrix that rotates every vector by angle $\theta$.

Question 15

1. $\begin{bmatrix} 4 \\ -9 \end{bmatrix}$ (correct answer)
2. $\begin{bmatrix} 6 \\ -6 \end{bmatrix}$
3. $\begin{bmatrix} 5 \\ -7 \end{bmatrix}$
4. $\begin{bmatrix} 1 \\ -5 \end{bmatrix}$

Explanation: To find $A\vec{v}$, we multiply: $\begin{bmatrix} 2 & 1 \\ -1 & 3 \end{bmatrix} \begin{bmatrix} 3 \\ -2 \end{bmatrix} = \begin{bmatrix} 2(3) + 1(-2) \\ -1(3) + 3(-2) \end{bmatrix} = \begin{bmatrix} 6 - 2 \\ -3 - 6 \end{bmatrix} = \begin{bmatrix} 4 \\ -9 \end{bmatrix}$

Question 16

1. The transformation collapses the entire plane onto a line or point, making it non-invertible and reducing dimensionality (correct answer)
2. The transformation preserves all areas in the plane exactly, neither expanding nor contracting any regions
3. The transformation is a pure rotation that preserves both distances and angles between all vectors in the plane
4. The transformation can only be applied to vectors lying on the coordinate axes and is undefined elsewhere

Explanation: When $\det(A) = 0$, the transformation is not invertible and maps the entire plane onto a lower-dimensional space (a line through the origin or just the origin itself). The transformation is degenerate and loses information.

Question 17

1. The unit square becomes a parallelogram with vertices at $(0,0)$, $(1,0)$, $(3,1)$, and $(2,1)$ (correct answer)
2. The unit square becomes a triangle with vertices at $(0,0)$, $(1,0)$, and $(2,1)$, losing one vertex
3. The unit square remains unchanged since the transformation matrix has ones on the main diagonal
4. The unit square becomes a rectangle with vertices at $(0,0)$, $(2,0)$, $(2,2)$, and $(0,2)$

Explanation: Applying $B = \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix}$ to the vertices: $(0,0) \mapsto (0,0)$, $(1,0) \mapsto (1,0)$, $(1,1) \mapsto (3,1)$, $(0,1) \mapsto (2,1)$. This creates a parallelogram through shearing.

Question 18

1. The transformation associated with matrix $\begin{bmatrix} 2 & 1 \\ 1 & -3 \end{bmatrix}$ applied to vector $\begin{bmatrix} x \\ y \end{bmatrix}$ (correct answer)
2. The transformation associated with matrix $\begin{bmatrix} 2 & 1 \\ -3 & 1 \end{bmatrix}$ applied to vector $\begin{bmatrix} x \\ y \end{bmatrix}$
3. The transformation associated with matrix $\begin{bmatrix} x & y \\ 2 & 1 \\ 1 & -3 \end{bmatrix}$ in standard position
4. The transformation associated with matrix $\begin{bmatrix} 1 & -3 \\ 2 & 1 \end{bmatrix}$ applied to vector $\begin{bmatrix} x \\ y \end{bmatrix}$

Explanation: The transformation $(x,y) \mapsto (2x+y, x-3y)$ corresponds to matrix multiplication $\begin{bmatrix} 2 & 1 \\ 1 & -3 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 2x+y \\ x-3y \end{bmatrix}$.

Question 19

1. The function $L$ satisfies $L(c\vec{v}) = cL(\vec{v})$ for any scalar $c$ and vector $\vec{v}$ in $\mathbb{R}^2$ (correct answer)
2. The function $L$ always increases the length of every non-zero vector $\vec{v}$ in its domain $\mathbb{R}^2$
3. The function $L$ preserves all angles between vectors regardless of the specific matrix $A$ being used
4. The function $L$ maps every vector $\vec{v}$ to a vector that is perpendicular to the original vector $\vec{v}$

Explanation: Linear transformations satisfy the property $L(c\vec{v}) = cL(\vec{v})$ for any scalar $c$ and vector $\vec{v}$. This is one of the defining properties of linearity, along with $L(\vec{u} + \vec{v}) = L(\vec{u}) + L(\vec{v})$.

Question 20

1. The vector is transformed to $\begin{bmatrix} b \\ a \end{bmatrix}$, which represents a reflection across the line $y = x$ (correct answer)
2. The vector is transformed to $\begin{bmatrix} -a \\ -b \end{bmatrix}$, which represents a rotation by $180°$ about the origin
3. The vector is transformed to $\begin{bmatrix} a \\ -b \end{bmatrix}$, which represents a reflection across the horizontal $x$-axis
4. The vector is transformed to $\begin{bmatrix} -b \\ a \end{bmatrix}$, which represents a rotation by $90°$ counterclockwise about the origin

Explanation: The matrix $\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$ applied to $\begin{bmatrix} a \\ b \end{bmatrix}$ gives $\begin{bmatrix} b \\ a \end{bmatrix}$, which swaps the coordinates. This represents a reflection across the line $y = x$.