Linear Transformations and Matrices - AP Precalculus

1. Which of the following best describes what a linear transformation is?
   Answer: 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 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.
2. How can a set of vectors in $\mathbb{R}^2$ be expressed for use in linear transformations?
   Answer: As a $2 \times n$ matrix where each column represents one of the $n$ vectors in the set 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.
3. If matrix $B = \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix}$, what geometric effect does this transformation have on the unit square with vertices at $(0,0)$, $(1,0)$, $(1,1)$, and $(0,1)$?
   Answer: The unit square becomes a parallelogram with vertices at $(0,0)$, $(1,0)$, $(3,1)$, and $(2,1)$ 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.
4. If $\vec{v} = \begin{bmatrix} 3 \\ -2 \end{bmatrix}$ and the linear transformation matrix is $A = \begin{bmatrix} 2 & 1 \\ -1 & 3 \end{bmatrix}$, what is $A\vec{v}$?
   Answer: $\begin{bmatrix} 4 \\ -9 \end{bmatrix}$ To find $A\vec{v}$, we multiply: $\begin{bmatrix} 2 & 1 \\ -1 & 3 \end{bmatrix} \begin{bmatrix} 3 \\ -2 \end{bmatrix} = \begin{bmatrix} 2(3) + 1(-2) \\ -1(3) + 3(-2) \end{bmatrix} = \begin{bmatrix} 6 - 2 \\ -3 - 6 \end{bmatrix} = \begin{bmatrix} 4 \\ -9 \end{bmatrix}$
5. What happens when the transformation matrix $\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$ is applied to a vector $\begin{bmatrix} a \\ b \end{bmatrix}$?
   Answer: The vector is transformed to $\begin{bmatrix} b \\ a \end{bmatrix}$, which represents a reflection across the line $y = x$ 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 $y = x$.
6. What happens when a linear transformation is applied to the zero vector?
   Answer: The result is always the zero vector regardless of the transformation matrix used 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.
7. For a linear transformation $L$ from $\mathbb{R}^2$ to $\mathbb{R}^2$, what is the relationship between $L$ and its associated matrix $A$?
   Answer: There exists a unique $2 \times 2$ matrix $A$ such that $L(\vec{v}) = A\vec{v}$ for all vectors $\vec{v}$ in $\mathbb{R}^2$ 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.
8. When multiplying a $2 \times 2$ transformation matrix $A$ by a $2 \times n$ matrix of input vectors, what is the result?
   Answer: A $2 \times n$ matrix containing the output vectors from applying transformation $A$ to each input vector 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.
9. If $A = \begin{bmatrix} 2 & 0 \\ 0 & 3 \end{bmatrix}$, what type of transformation does matrix $A$ represent?
   Answer: A scaling transformation that stretches vectors by factor $2$ in the $x$-direction and factor $3$ in the $y$-direction 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.
10. A design tool reflects vectors across the line $y=x$ using $M=[0\ 1;\ 1\ 0]$. Based on the scenario, what is $M\langle -2,5\rangle$?
    Answer: $\langle 5,-2\rangle$ This question tests AP Precalculus skills, specifically linear transformations and matrices. A reflection matrix across the line y=x has the form M=[0 1; 1 0], which swaps the x and y coordinates of any point. In this scenario, the transformation matrix M is applied to vector ⟨-2,5⟩, demonstrating how reflection across y=x exchanges coordinate positions. Choice B is correct because M⟨-2,5⟩ = [0 1; 1 0]⟨-2,5⟩ = ⟨0(-2)+1(5), 1(-2)+0(5)⟩ = ⟨5,-2⟩, showing that the coordinates are swapped from (-2,5) to (5,-2). Choice C is incorrect because it negates the x-coordinate after swapping, which is not part of the reflection across y=x. To help students: Remember that reflection across y=x simply exchanges x and y coordinates. Visualize this as folding the plane along the line y=x.
11. For the linear transformation given by $L(\vec{v}) = A\vec{v}$ where $A$ is a $2 \times 2$ matrix, which statement is always true?
    Answer: The function $L$ satisfies $L(c\vec{v}) = cL(\vec{v})$ for any scalar $c$ and vector $\vec{v}$ in $\mathbb{R}^2$ 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 $L(\vec{u} + \vec{v}) = L(\vec{u}) + L(\vec{v})$.
12. If $A = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$ is a transformation matrix, what geometric transformation does it represent?
    Answer: A reflection across the $x$-axis that changes the sign of the $y$-coordinate while preserving the $x$-coordinate 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.
13. If a $2 \times 2$ matrix $A$ represents a linear transformation and $\det(A) = 0$, what does this tell us about the transformation?
    Answer: The transformation collapses the entire plane onto a line or point, making it non-invertible and reducing dimensionality 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.
14. A camera effect applies vertical shear using $V=[1\ 0;\ -3\ 1]$. Based on the scenario, what is $V\langle 2,1\rangle$?
    Answer: $\langle 2,-5\rangle$ This question tests AP Precalculus skills, specifically linear transformations and matrices. A vertical shear matrix has the form V=[1 0; k 1], where k is the shear factor that adds k times the x-coordinate to the y-coordinate. In this scenario, the transformation matrix V=[1 0; -3 1] applies a vertical shear with factor -3 to vector ⟨2,1⟩. Choice A is correct because V⟨2,1⟩ = [1 0; -3 1]⟨2,1⟩ = ⟨1(2)+0(1), -3(2)+1(1)⟩ = ⟨2, -6+1⟩ = ⟨2,-5⟩, showing that the x-coordinate remains 2 while the y-coordinate becomes -3(2)+1=-5. Choice C is incorrect because it suggests a positive y-coordinate, failing to account for the negative shear factor. To help students: Remember that vertical shear keeps x-coordinates fixed while modifying y-coordinates based on the x-value. Pay attention to the sign of the shear factor.
15. A graphics filter shears horizontally using $H=[1\ 2;\ 0\ 1]$. Using the given transformation, what is $H\langle 1,3\rangle$?
    Answer: $\langle 7,3\rangle$ This question tests AP Precalculus skills, specifically linear transformations and matrices. A horizontal shear matrix has the form H=[1 k; 0 1], where k is the shear factor that adds k times the y-coordinate to the x-coordinate. In this scenario, the transformation matrix H=[1 2; 0 1] applies a horizontal shear with factor 2 to vector ⟨1,3⟩. Choice A is correct because H⟨1,3⟩ = [1 2; 0 1]⟨1,3⟩ = ⟨1(1)+2(3), 0(1)+1(3)⟩ = ⟨1+6, 3⟩ = ⟨7,3⟩, showing that the x-coordinate becomes 1+2(3)=7 while the y-coordinate remains 3. Choice B is incorrect because it suggests minimal change to the coordinates, failing to apply the shear transformation properly. To help students: Remember that horizontal shear keeps y-coordinates fixed while modifying x-coordinates based on the y-value. Visualize shearing as slanting a shape horizontally.
16. A photo editor scales a logo using $S=[2\ 0;\ 0\ 3]$. Based on the scenario, what changes occur to the dimensions?
    Answer: Width doubles; height triples. This question tests AP Precalculus skills, specifically linear transformations and matrices. A scaling matrix of the form S=[a 0; 0 b] scales the x-coordinate by factor a and the y-coordinate by factor b. In this scenario, the transformation matrix S=[2 0; 0 3] is applied to scale a logo, where the diagonal entries 2 and 3 represent the scaling factors for width and height respectively. Choice A is correct because the matrix multiplies the x-coordinate (width) by 2 and the y-coordinate (height) by 3, resulting in the width doubling and height tripling. Choice B is incorrect because it reverses the scaling factors, misinterpreting which diagonal entry affects which dimension. To help students: Emphasize that in a diagonal scaling matrix, the first diagonal entry scales the x-coordinate (width) and the second scales the y-coordinate (height). Practice identifying scaling factors directly from the matrix entries.
17. A navigation system reflects a heading across the $x$-axis using $M=[1\ 0;\ 0\ -1]$. Using the given transformation, what is $M\langle 4,-3\rangle$?
    Answer: $\langle 4,3\rangle$ This question tests AP Precalculus skills, specifically linear transformations and matrices. A reflection matrix across the x-axis has the form M=[1 0; 0 -1], which keeps the x-coordinate unchanged while negating the y-coordinate. In this scenario, the transformation matrix M is applied to vector ⟨4,-3⟩, demonstrating how reflection across the x-axis affects a point's position. Choice B is correct because M⟨4,-3⟩ = [1 0; 0 -1]⟨4,-3⟩ = ⟨1(4)+0(-3), 0(4)+(-1)(-3)⟩ = ⟨4,3⟩, showing that the x-coordinate remains 4 while the y-coordinate changes from -3 to 3. Choice A is incorrect because it negates both coordinates, which would represent a reflection through the origin rather than across the x-axis. To help students: Remember that reflection across the x-axis only changes the sign of the y-coordinate. Visualize the geometric transformation to verify algebraic results.
18. If transformation matrix $A$ has determinant $-6$, what can be concluded about the transformation?
    Answer: The transformation changes areas by a factor of $6$ and reverses orientation of regions in the plane 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.
19. What information do the unit vectors provide when determining the matrix associated with a linear transformation?
    Answer: The images of the unit vectors under the transformation become the columns of the transformation matrix 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 $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$.
20. What is the result of applying the linear transformation matrix $\begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}$ to the vector $\begin{bmatrix} 4 \\ -3 \end{bmatrix}$?
    Answer: $\begin{bmatrix} -4 \\ -3 \end{bmatrix}$, representing a reflection of the original vector across the $y$-axis $\begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 4 \\ -3 \end{bmatrix} = \begin{bmatrix} -4 \\ -3 \end{bmatrix}$. This matrix reflects across the $y$-axis, negating the $x$-coordinate while keeping the $y$-coordinate unchanged.

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