Practice Test 2 - Linear Algebra

20 questions. Answer them in any order, then submit to see your score and every explanation.

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1. Let $\vec{a} = \langle 1, 1, 0 \rangle$ and $\vec{b} = \langle 0, 1, 1 \rangle$. Let $\vec{u} = \vec{a} + \vec{b}$ and $\vec{v} = \vec{a} - \vec{b}$. What is the cosine of the angle $\theta$ between $\vec{u}$ and $\vec{v}$?
2. Let $A$ be an $m \times n$ matrix and $\mathbf{b}$ be a vector in $\mathbb{R}^m$. If $\hat{\mathbf{x}}$ is the least-squares solution to the system $A\mathbf{x} = \mathbf{b}$, which statement best describes the geometric relationship involving the error vector $\mathbf{e} = \mathbf{b} - A\hat{\mathbf{x}}$?
3. What is the geometric interpretation of the inverse of the transformation represented by the matrix $A = \begin{pmatrix} 1 & -1 \\ 0 & 1 \end{pmatrix}$?
4. Let $T: \mathbb{R}^3 \to \mathbb{R}^3$ be a linear transformation with matrix representation $A = \begin{pmatrix} 2 & 1 & 0 \\ 0 & 2 & 1 \\ 0 & 0 & 2 \end{pmatrix}$. If $B = \{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ is a basis for $\mathbb{R}^3$ such that $[T]_B$ is diagonal, which condition must the basis vectors satisfy?
5. A symmetric matrix $A$ is orthogonally diagonalized as $A = PDP^T$ with $P = \begin{pmatrix} 1/\sqrt{5} & -2/\sqrt{5} \\ 2/\sqrt{5} & 1/\sqrt{5} \end{pmatrix}$ and $D = \begin{pmatrix} 10 & 0 \\ 0 & 5 \end{pmatrix}$. What is the matrix $A$?
6. Let $A$ be a $5 \times 8$ matrix such that the solution set to the homogeneous system $A\mathbf{x} = \mathbf{0}$ can be described with 4 free parameters. What is the dimension of the column space of $A$?
7. A student is solving for $x$ in the system $\begin{cases} 3x + y = 5 \\ 2x - 4y = 1 \end{cases}$ using Cramer's rule. Their work is shown below. Step 1: $D = \det \begin{pmatrix} 3 & 1 \\ 2 & -4 \end{pmatrix} = -12 - 2 = -14$. Step 2: $D_x = \det \begin{pmatrix} 3 & 5 \\ 2 & 1 \end{pmatrix} = 3 - 10 = -7$. Step 3: $x = \frac{D_x}{D} = \frac{-7}{-14} = \frac{1}{2}$. In which step did the student make their first mistake?
8. A spring-mass system with damping leads to the equation $m\ddot{x} + c\dot{x} + kx = 0$. When written as a first-order system $\mathbf{y}' = A\mathbf{y}$ with $\mathbf{y} = \begin{pmatrix} x \\ \dot{x} \end{pmatrix}$, under what condition on the parameters will the system be critically damped (repeated real eigenvalues)?
9. A student performs the row operation $R_2 \leftarrow R_2 + 3R_1$ on $\begin{bmatrix} 2 & -1 & 4 & | & 5 \\ -6 & 3 & -12 & | & -15 \\ 1 & 0 & 2 & | & 3 \end{bmatrix}$ and obtains $\begin{bmatrix} 2 & -1 & 4 & | & 5 \\ 0 & 0 & 0 & | & 0 \\ 1 & 0 & 2 & | & 3 \end{bmatrix}$. The student then concludes that the system has infinitely many solutions because the second row became all zeros. What should the student do next to properly analyze this system?
10. Consider linear transformations $A: \mathbb{R}^n \to \mathbb{R}^n$ and $B: \mathbb{R}^n \to \mathbb{R}^n$ where $A^2 = A$ ($A$ is idempotent) and $B^2 = I$ ($B$ is involutory). If $C = B \circ A \circ B$, which property must $C$ satisfy?
11. Let $V$ and $W$ be vector spaces. A function $T: V \to W$ is a linear transformation if it satisfies additivity and homogeneity. Which of the following conditions is a valid, alternative, single-property definition for linearity?
12. For the overdetermined linear system $A\mathbf{x} = \mathbf{b}$ where $A$ is $m \times n$ with $m > n$ and $\text{rank}(A) = n$, consider the augmented matrix $[A | \mathbf{b}]$. If the least squares residual has norm $r > 0$, which statement about the rank of $[A | \mathbf{b}]$ is correct?
13. A $3 \times 3$ matrix $A$ is constructed using the rule $a_{ij} = 2i + 2j$ for its entries. Which of the following properties does matrix $A$ have?
14. A shear transformation in $\mathbb{R}^2$ is followed by a rotation of $90°$ counterclockwise. If the original parallelogram had area 12 and the final parallelogram has area 36, what is the determinant of the shear transformation matrix?
15. During the process of finding the inverse of a matrix $A$ by applying row operations to $[A|I]$, the augmented matrix is reduced to $\begin{pmatrix} 1 & -2 & | & 1 & 0 \\ 0 & 0 & | & 3 & 1 \end{pmatrix}$. What can be concluded about the inverse of matrix $A$?
16. A $3 \times 3$ symmetric matrix $A$ has eigenvalues $\lambda_1 = 4$, $\lambda_2 = 1$, and $\lambda_3 = -2$. If the eigenvector corresponding to $\lambda_1 = 4$ is $\mathbf{v}_1 = \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}$, and after applying the Gram-Schmidt process to obtain an orthonormal basis, the first column of the orthogonal matrix $P$ is $\begin{pmatrix} 1/\sqrt{6} \\ 2/\sqrt{6} \\ 1/\sqrt{6} \end{pmatrix}$. What is the trace of $A^2$?
17. Let $A$ be a $3 \times 3$ matrix. The matrix $B = A + A^T$ is given by $B = \begin{pmatrix} 4 & 6 & 0 \\ 6 & -2 & 5 \\ 0 & 5 & 8 \end{pmatrix}$. Based on this information, what are the diagonal entries of matrix $A$?
18. A $3 \times 3$ matrix $A$ has the property that $\det(A) = -8$. If $B = 2A^T$, where $A^T$ denotes the transpose of $A$, what is the relationship between the orientations induced by transformations $A$ and $B$?
19. Consider a $4 \times 4$ matrix $R$ with characteristic polynomial $p(x) = (x-1)^2(x-3)^2$. If $R$ is diagonalizable and satisfies $R^2 - 4R + 3I = S$ for some matrix $S$, what is the rank of $S$?
20. If an $n \times n$ matrix $A$ has rank $n$, which of the following is a necessary consequence?

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