This question tests understanding of how matrices transform vectors through matrix-vector multiplication. Geometric transformations like rotations, reflections, and scalings can be represented by matrices: applying the matrix to a vector transforms that vector according to the geometric operation the matrix encodes. The matrix $\begin{pmatrix}-1 & 0 \\ 0 & 1\end{pmatrix}$ reflects vectors across the y-axis, so applying it to vector $\begin{pmatrix}4 \\ -2\end{pmatrix}$ transforms it to $\begin{pmatrix}-4 \\ -2\end{pmatrix}$, which represents the point reflected across the y-axis. Choice C is correct because it accurately performs the matrix-vector multiplication. Choice A returns the original vector unchanged, failing to apply the transformation at all. Common 2×2 transformations to recognize: $\begin{pmatrix}1 & 0 \\ 0 & -1\end{pmatrix}$ reflects across x-axis, $\begin{pmatrix}0 & -1 \\ 1 & 0\end{pmatrix}$ rotates 90° counterclockwise, $\begin{pmatrix}k & 0 \\ 0 & k\end{pmatrix}$ scales uniformly by k, and $\begin{pmatrix}1 & 0 \\ 0 & 1\end{pmatrix}$ is the identity (no change). Remember that matrix transformations are linear: if you understand what the matrix does to the basis vectors (like $\begin{pmatrix}1 \\ 0\end{pmatrix}$ and $\begin{pmatrix}0 \\ 1\end{pmatrix}$), you can determine what it does to any vector by using linearity.

This question tests understanding of how matrices transform vectors through matrix-vector multiplication. When a matrix A multiplies a vector v (written as Av), the result is a new vector found by taking the dot product of each row of A with the column vector v: for a $2 \times 2$ matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ and vector $\begin{pmatrix} x \\ y \end{pmatrix}$, the result is $\begin{pmatrix} ax + by \\ cx + dy \end{pmatrix}$. For arbitrary vector $\begin{pmatrix} x \\ y \end{pmatrix}$, the transformation by matrix $\begin{pmatrix} 2 & 1 \\ -3 & 0 \end{pmatrix}$ gives $\begin{pmatrix} 2x + y \\ -3x + 0y \end{pmatrix}$, showing how each component of the result depends on both components of the input through the matrix entries. Choice A is correct because it properly computes the dot products for each row. Choice B makes an error in computing the second component, calculating -3y instead of -3x using dot product. Key to matrix-vector multiplication: treat the vector as a column matrix and compute each entry of the result as the dot product of the corresponding row of the matrix with the vector column—for row 1: $a \cdot x + b \cdot y$, for row 2: $c \cdot x + d \cdot y$. To check your work, verify dimensions: a $2 \times 2$ matrix times a $2 \times 1$ vector must give a $2 \times 1$ vector, and each component should come from one complete dot product calculation.

This question tests understanding of how matrices transform vectors through matrix-vector multiplication. When a matrix A multiplies a vector v (written as Av), the result is a new vector found by taking the dot product of each row of A with the column vector v: for a $2 \times 2$ matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ and vector $\begin{pmatrix} x \\ y \end{pmatrix}$, the result is $\begin{pmatrix} a x + b y \\ c x + d y \end{pmatrix}$. For matrix A = $\begin{pmatrix} 3 & 0 \\ -1 & 2 \end{pmatrix}$ and vector v = $\begin{pmatrix} 1 \\ 4 \end{pmatrix}$, we compute Av by taking the dot product of each row with v. First entry: $3 \cdot 1 + 0 \cdot 4 = 3 + 0 = 3$. Second entry: $(-1) \cdot 1 + 2 \cdot 4 = -1 + 8 = 7$. Thus Av = $\begin{pmatrix} 3 \\ 7 \end{pmatrix}$. Choice A is correct because it properly computes the dot products for each row. Choice B makes an error in computing the first component, calculating $3*4$ or something to get 12 instead of 3. Key to matrix-vector multiplication: treat the vector as a column matrix and compute each entry of the result as the dot product of the corresponding row of the matrix with the vector column—for row 1: $a \cdot x + b \cdot y$, for row 2: $c \cdot x + d \cdot y$. To check your work, verify dimensions: a $2 \times 2$ matrix times a $2 \times 1$ vector must give a $2 \times 1$ vector, and each component should come from one complete dot product calculation.

This question tests understanding of how matrices transform vectors through matrix-vector multiplication. Geometric transformations like rotations, reflections, and scalings can be represented by matrices: applying the matrix to a vector transforms that vector according to the geometric operation the matrix encodes. The matrix $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ rotates vectors 90° counterclockwise, so applying it to vector $\begin{pmatrix} 2 \\ -3 \end{pmatrix}$ transforms it to $\begin{pmatrix} 3 \\ 2 \end{pmatrix}$, which represents the point rotated 90°. Choice A is correct because it accurately performs the matrix-vector multiplication. Choice B makes an arithmetic error, computing something like $\begin{pmatrix} -1 \times 2 + 0 \times(-3) \\ 0 \times 2 + (-1) \times(-3) \end{pmatrix}$ but that's not accurate; it might confuse the matrix entries. Common 2×2 transformations to recognize: $\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$ reflects across x-axis, $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ rotates 90° counterclockwise, $\begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix}$ scales uniformly by k, and $\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ is the identity (no change). Remember that matrix transformations are linear: if you understand what the matrix does to the basis vectors (like $\begin{pmatrix} 1 \\ 0 \end{pmatrix}$ and $\begin{pmatrix} 0 \\ 1 \end{pmatrix}$), you can determine what it does to any vector by using linearity.

This question tests understanding of how matrices transform vectors through matrix-vector multiplication. Geometric transformations like rotations, reflections, and scalings can be represented by matrices: applying the matrix to a vector transforms that vector according to the geometric operation the matrix encodes. For matrix $H = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}$ and vector $\mathbf{v} = \begin{pmatrix} 1 \\ 3 \end{pmatrix}$, we compute $Hv$ by taking the dot product of each row with $v$. First entry: $1 \cdot 1 + 2 \cdot 3 = 1 + 6 = 7$. Second entry: $0 \cdot 1 + 1 \cdot 3 = 0 + 3 = 3$. Thus $Hv = \begin{pmatrix} 7 \\ 3 \end{pmatrix}$. Choice B is correct because it properly computes the dot products for each row. Choice A makes an error in computing the first component, calculating $1+2 \cdot 3$ but perhaps adding wrong to get 6 or confusing. Key to matrix-vector multiplication: treat the vector as a column matrix and compute each entry of the result as the dot product of the corresponding row of the matrix with the vector column—for row 1: $a \cdot x + b \cdot y$, for row 2: $c \cdot x + d \cdot y$. To check your work, verify dimensions: a $2\times2$ matrix times a $2\times1$ vector must give a $2\times1$ vector, and each component should come from one complete dot product calculation.

This question tests understanding of how matrices transform vectors through matrix-vector multiplication. Geometric transformations like rotations, reflections, and scalings can be represented by matrices: applying the matrix to a vector transforms that vector according to the geometric operation the matrix encodes. The matrix $$ \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} $$ reflects vectors across the x-axis, so applying it to vector $$ \begin{pmatrix} -1 \\ 4 \end{pmatrix} $$ transforms it to $$ \begin{pmatrix} -1 \\ -4 \end{pmatrix} $$, which represents the point reflected across the x-axis. Choice C is correct because it accurately performs the matrix-vector multiplication. Choice B has the correct components but wrong sign in the second, perhaps failing to apply the -1 scalar to the y-component. Common $2 \times 2$ transformations to recognize: $$ \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} $$ reflects across x-axis, $$ \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} $$ rotates 90° counterclockwise, $$ \begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix} $$ scales uniformly by k, and $$ \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} $$ is the identity (no change). Remember that matrix transformations are linear: if you understand what the matrix does to the basis vectors (like $$ \begin{pmatrix} 1 \\ 0 \end{pmatrix} $$ and $$ \begin{pmatrix} 0 \\ 1 \end{pmatrix} $$), you can determine what it does to any vector by using linearity.

This question tests understanding of how matrices transform vectors through matrix-vector multiplication. Geometric transformations like rotations, reflections, and scalings can be represented by matrices: applying the matrix to a vector transforms that vector according to the geometric operation the matrix encodes. The matrix $\begin{pmatrix} 2 & 0 \\ 0 & 1 \end{pmatrix}$ scales the x-component by 2 while leaving y unchanged, so applying it to vector $\begin{pmatrix} 3 \\ -2 \end{pmatrix}$ transforms it to $\begin{pmatrix} 6 \\ -2 \end{pmatrix}$, which represents the point scaled along the x-axis. Choice A is correct because it accurately performs the matrix-vector multiplication. Choice B makes an error in computing the first component, calculating 3 instead of 6, perhaps forgetting to multiply by 2. Common 2×2 transformations to recognize: $\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$ reflects across x-axis, $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ rotates 90° counterclockwise, $\begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix}$ scales uniformly by k, and $\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ is the identity (no change). Remember that matrix transformations are linear: if you understand what the matrix does to the basis vectors (like $\begin{pmatrix} 1 \\ 0 \end{pmatrix}$ and $\begin{pmatrix} 0 \\ 1 \end{pmatrix}$), you can determine what it does to any vector by using linearity.

This question tests understanding of how matrices transform vectors through matrix-vector multiplication. Geometric transformations like rotations, reflections, and scalings can be represented by matrices: applying the matrix to a vector transforms that vector according to the geometric operation the matrix encodes. The matrix [1 1; 0 1] represents a horizontal shear, transforming [x; y] to [x + y; y], which slides points horizontally based on their y-coordinate. Choice C is correct because it correctly identifies the geometric transformation. Choice B incorrectly describes the geometric transformation, claiming it is a 90° clockwise rotation when actually it is a shear. Common 2×2 transformations to recognize: [1 0; 0 -1] reflects across x-axis, [0 -1; 1 0] rotates 90° counterclockwise, [k 0; 0 k] scales uniformly by k, and [1 0; 0 1] is the identity (no change). Remember that matrix transformations are linear: if you understand what the matrix does to the basis vectors (like [1; 0] and [0; 1]), you can determine what it does to any vector by using linearity.

This question tests understanding of how matrices transform vectors through matrix-vector multiplication. When a matrix A multiplies a vector v (written as Av), the result is a new vector found by taking the dot product of each row of A with the column vector v: for a 2×2 matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ and vector $\begin{bmatrix} x \\ y \end{bmatrix}$, the result is $\begin{bmatrix} ax+by \\ cx+dy \end{bmatrix}$. For arbitrary vector $\begin{bmatrix} x \\ y \end{bmatrix}$, the transformation by matrix $\begin{bmatrix} 3 & 0 \\ 1 & -2 \end{bmatrix}$ gives $\begin{bmatrix} 3x+0y \\ 1x-2y \end{bmatrix}$, showing how each component of the result depends on both components of the input through the matrix entries. Choice A is correct because it properly computes the dot products for each row. Choice B makes an error in computing the second component, using y instead of x for the first term. Key to matrix-vector multiplication: treat the vector as a column matrix and compute each entry of the result as the dot product of the corresponding row of the matrix with the vector column—for row 1: $a \cdot x + b \cdot y$, for row 2: $c \cdot x + d \cdot y$. To check your work, verify dimensions: a 2×2 matrix times a 2×1 vector must give a 2×1 vector, and each component should come from one complete dot product calculation.

This question tests understanding of how matrices transform vectors through matrix-vector multiplication. Geometric transformations like rotations, reflections, and scalings can be represented by matrices: applying the matrix to a vector transforms that vector according to the geometric operation the matrix encodes. The matrix $\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$ rotates vectors 90° counterclockwise, so applying it to vector $\begin{bmatrix} 4 \\ -1 \end{bmatrix}$ transforms it to $\begin{bmatrix} -(-1) \\ 4 \end{bmatrix} = \begin{bmatrix} 1 \\ 4 \end{bmatrix}$, which represents the point rotated 90°. Choice C is correct because it accurately performs the matrix-vector multiplication. Choice D has the correct components but in reversed order, giving $\begin{bmatrix} 4 \\ -1 \end{bmatrix}$ rotated clockwise instead of counterclockwise. Common 2×2 transformations to recognize: $\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$ reflects across x-axis, $\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$ rotates 90° counterclockwise, $\begin{bmatrix} k & 0 \\ 0 & k \end{bmatrix}$ scales uniformly by k, and $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ is the identity (no change). To check your work, verify dimensions: a $2 \times 2$ matrix times a $2 \times 1$ vector must give a $2 \times 1$ vector, and each component should come from one complete dot product calculation.