Practice Test 4 - Linear Algebra

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

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1. Consider the system of linear equations $Ax = b$ where $A$ is a $4 \times 5$ matrix with rank 3. If the system is consistent and has the parametric solution $x = \begin{pmatrix} 2 \\ -1 \\ 0 \\ 3 \\ 0 \end{pmatrix} + t\begin{pmatrix} 1 \\ 2 \\ -1 \\ 0 \\ 1 \end{pmatrix} + s\begin{pmatrix} 0 \\ 1 \\ 1 \\ -2 \\ 3 \end{pmatrix}$, which statement best describes the solution set?
2. Matrix $F = \begin{pmatrix} k & 2 & -1 \\ 1 & k & 2 \\ -1 & 1 & k \end{pmatrix}$ has the property that $\det(F) = 0$ for exactly three distinct values of $k$. What is the sum of these three values?
3. For the quadratic form $Q(x,y,z) = 2x^2 + 3y^2 + z^2 + 4xy - 2xz + 6yz$, suppose we want to eliminate all cross-product terms by rotating to principal axes. If the coefficient matrix has eigenvalues $\lambda_1 = 6$, $\lambda_2 = 3$, and $\lambda_3 = -3$, what type of quadric surface does the equation $Q(x,y,z) = 18$ represent?
4. Consider the matrix $A = \begin{pmatrix} 3 & 0 & 0 \\ -1 & 2 & 0 \\ 4 & 1 & k \end{pmatrix}$. The sum of the eigenvalues of the matrix $A^2$ is 29. What is the value of $k^2$?
5. A mechanical system satisfies $\mathbf{x}_{k+1} = M\mathbf{x}_k$ where $M$ has characteristic polynomial $p(\lambda) = \lambda^2 - 0.6\lambda + 0.25$. What can be determined about the stability without computing eigenvectors?
6. Let $A = \begin{pmatrix} 2 & 5 \\ -1 & -3 \end{pmatrix}$. The product $A\mathbf{x}$ results in the vector $\mathbf{b} = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$. What is vector $\mathbf{x}$?
7. The general real solution to a $2 \times 2$ linear system $\mathbf{x}' = A\mathbf{x}$ is given by $\mathbf{x}(t) = c_1 e^{3t} \begin{pmatrix} \cos(4t) \\ -2\sin(4t) \end{pmatrix} + c_2 e^{3t} \begin{pmatrix} \sin(4t) \\ 2\cos(4t) \end{pmatrix}$. What are the eigenvalues of the matrix $A$?
8. For the matrix $A = \begin{pmatrix} \cos(k) & -\sin(k) & 0 \\ \sin(k) & \cos(k) & 0 \\ 0 & 0 & k \end{pmatrix}$ to be an orthogonal matrix, the value of $k$ must be:
9. Consider a $5 \times 4$ matrix $J$ where every vector in $\mathbb{R}^4$ can be written uniquely as $\mathbf{v} + \mathbf{w}$ where $\mathbf{v} \in$ null($J$) and $J\mathbf{w} \in$ column($J$). What must be true about the rank of $J$?
10. Suppose a linear transformation $T: \mathbb{R}^2 \to \mathbb{R}^2$ is represented by a matrix $A$ that is diagonalizable. If the eigenvalues of $A$ are $\lambda_1 = 1$ and $\lambda_2 = 0$, what is the geometric interpretation of the transformation $T$?
11. Let $S$ be a real symmetric $5 \times 5$ matrix with eigenvalues $-2, -1, 0, 1, 3$. If $T = S^4 + 2S^2$, which statement about the definiteness of $T$ is correct?
12. The solution set of a homogeneous system $\mathbf{A}\vec{x} = \vec{0}$ is given by all linear combinations of the vectors $\vec{u} = \begin{pmatrix} 1 \\ 0 \\ -2 \\ 1 \end{pmatrix}$ and $\vec{v} = \begin{pmatrix} 0 \\ 1 \\ 3 \\ 0 \end{pmatrix}$. Which of the following vectors is NOT a solution to this system?
13. Let $T: \mathbb{R}^2 \to \mathbb{R}^2$ be a horizontal shear that maps the standard basis vector $\mathbf{e}_2$ to $\mathbf{e}_2 - 2\mathbf{e}_1$ and leaves $\mathbf{e}_1$ fixed. What is the standard matrix for the transformation $T^3 = T \circ T \circ T$?
14. A matrix $A$ is transformed into a matrix $B$ by the following sequence of operations on $A$'s rows: 1. First, scale row 2 by a factor of 3. 2. Second, add the new row 2 to row 1. Which single matrix $P$ satisfies the equation $PA=B$?
15. Let $W$ be a subspace of $\mathbb{R}^m$ and let $\mathbf{b}$ be a vector in $\mathbb{R}^m$. The least-squares solution $\hat{\mathbf{x}}$ to $A\mathbf{x}=\mathbf{b}$ (where $W=\mathrm{Col}(A)$) minimizes the distance $\|\mathbf{b} - \mathbf{w}\|$ for all $\mathbf{w} \in W$. Which expression represents the square of this minimum distance?
16. The system of linear equations $\begin{cases} 3x_1 - 2x_2 = 5 \\ x_1 + 4x_2 = 7 \end{cases}$ can be written in the matrix form $A\mathbf{x} = \mathbf{b}$. How can the vector $\mathbf{b}$ be expressed as a linear combination of the columns of $A$?
17. Consider the system of equations: $x + y - z = 2$ $x + 2y + z = 3$ $x + y + (c^2 - 5)z = c$ For which value of the constant $c$ will the system have no unique solution?
18. Given a symmetric matrix $A$, a student performs the following steps to find an orthogonal matrix $P$ for diagonalization: \\ 1. Find the distinct eigenvalues $\lambda_i$. \\ 2. For each $\lambda_i$, find a basis for the eigenspace $E_{\lambda_i}$. \\ 3. Form $P$ by placing the basis vectors found in step 2 as columns. \\ The student finds their matrix $P$ is not orthogonal. Which of the following describes the critical omission in their procedure?
19. Let $\vec{u}$ and $\vec{v}$ be non-zero vectors in $\mathbb{R}^n$. If the vector $\vec{u} + \vec{v}$ is orthogonal to the vector $\vec{u} - \vec{v}$, what must be true about $\vec{u}$ and $\vec{v}$?
20. Which of the following describes an impossible scenario for a $5 \times 8$ matrix $A$?

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