108 年 國立臺灣聯合大學系統（清華、政治、陽明交通、中央四校聯招）研究所電機類《工程數學（B）》

Consider a linear system

Which of the following statements is/are true?
(A) When $h = 12$ and $k = 9$, the system is inconsistent.
(B) When $h = 12$ and $k = 9$, the system has many solutions.
(C) When $h = 12$ and $k \neq 9$, the system is consistent.
(D) When $h = 12$ and $k \neq 9$, the system has at least one solution.
(E) When $h \neq 12$, the system has a unique solution.

Denote det A as the determinant of the matrix A, and denote $A^{-1}$ as the inverse of the matrix A. Let A, B, and P be square matrices. Which of the following statements is/are true?
(A) It is always true that $\det AB = \det BA$.
(B) If the columns of A are linearly dependent, then $\det A = 0$.
(C) It is always true that $\det (A + B) = \det A + \det B$.
(D) If A is invertible, then $\det A^{-1} = \frac{1}{\det A}$.
(E) Suppose that P is invertible. Then $\det (PAP^{-1}) = \det A$.

Let $A = \begin{bmatrix} 1 & -3 \\ 3 & 5 \\ -1 & 7 \end{bmatrix}$ and define a transformation $T: \mathbb{R}^2 \rightarrow \mathbb{R}^3$ by $T(x) = Ax$. Which of the following statements is/are true?
(A) The image of $x = \begin{bmatrix} 1.5 \\ -0.5 \end{bmatrix}$ under T is $\begin{bmatrix} 3 \\ 2 \\ -5 \end{bmatrix}$.
(B) There is exactly one $x$ whose image under T is $\begin{bmatrix} 3 \\ 2 \\ -5 \end{bmatrix}$.
(C) The vector $b$ is in the range of $T$ if $b$ is the image of some $x$ in $\mathbb{R}^2$.
(D) The vector $b$ is in the range of $T$ if the system $Ax = b$ is inconsistent.
(E) The vector $\begin{bmatrix} 3 \\ 2 \\ -5 \end{bmatrix}$ is not in the range of $T$.

The set $P_n$ of polynomials of degree at most $n$, $n \geq 0$, consists of all polynomials of the form $p(t) = a_0 + a_1t + a_2t^2 + \dots + a_nt^n$. Let $p_1(t) = 2 + 2t^2$, $p_2(t) = -t + 3t^2$, and $p_3(t) = 1 + t - 3t^2$. Which of the following statements is/are true?
(A) $p_1(t)$ is in $P_3$.
(B) To see whether the polynomials $p_1(t)$, $p_2(t)$, and $p_3(t)$ form a basis for $P_2$, we can place the coordinate vectors of the polynomials into the columns of a matrix and reduce the matrix to echelon form. If the resulting matrix is not invertible, then the polynomials $p_1(t)$, $p_2(t)$, and $p_3(t)$ form a basis for $P_2$.
(C) Polynomials $p_1(t)$, $p_2(t)$, and $p_3(t)$ form a basis for $P_2$.
(D) Consider the basis $B = \{p_1, p_2, p_3\}$ for $P_2$. Let $[q]_B$ be the B-coordinate vector of $q$. Given that $[q]_B = \begin{bmatrix} 2 \\ -1 \\ 1 \end{bmatrix}$, $q$ in $P_2$ is $2p_1 - p_2 + p_3$.
(E) Following (D), $q(t) = 2 - t + t^2$.

Suppose that a $6 \times 3$ matrix $A$ has rank 3. Denote $\dim H$ as the dimension of a nonzero subspace $H$, Nul $A$ as the null space of the matrix $A$, Row $A$ as the column space of the matrix $A$, rank $A$ as the rank of the matrix $A$, and $A^T$ as the transpose of the matrix $A$. Which of the following statements is/are true?
(A) $\dim$ Row $A = \text{rank } A$.
(B) $\dim$ Row $A = \text{rank } A^T$.
(C) $\dim$ Nul $A = 3$.
(D) $\dim$ Row $A = 6$.
(E) $\text{rank } A^T = 3$.

Let $y = \begin{bmatrix} 7 \\ 1 \\ 2 \\ -1 \end{bmatrix}$, $v_1 = \begin{bmatrix} 1 \\ 1 \\ 0 \\ 3 \end{bmatrix}$, and $v_2 = \begin{bmatrix} 1 \\ -1 \\ 1 \\ 0 \end{bmatrix}$. Let the subspace $W$ be spanned by $v_1$ and $v_2$. Which of the following statements is/are true?
(A) $v_1$ and $v_2$ are orthogonal.
(B) The closest point to $y$ in the subspace $W$ spanned by $v_1$ and $v_2$ is $\begin{bmatrix} 5 \\ 3 \\ 1 \\ 1 \end{bmatrix}$.
(C) The closest point in $W$ to $y$ is the projection of $y$ on $W$.
(D) The distance from the point $y$ in $\mathbb{R}^4$ to $W$ is defined as the distance from $y$ to the closest point in $W$.
(E) The distance from $y$ to the subspace of $\mathbb{R}^4$ spanned by $v_1$ and $v_2$ is 16.

Let $A^{-1}$ be the inverse of the matrix $A$, $A^T$ be the transpose of the matrix $A$, and $I$ be the identity matrix. Which of the following statements about invertible matrices is/are true?
(A) If $A$ is both diagonalizable and invertible, then so is $A^{-1}$.
(B) Suppose that $A = QR$, where $R$ is an invertible matrix. Then $A$ and $Q$ have the same column space.
(C) Suppose that $A$ and $B$ are square matrices, $B$ is invertible, and $AB$ is invertible. Then $A$ is invertible.
(D) Suppose that $A = PDQ^T$, where $P$ and $Q$ are $n \times n$ matrices with the property that $P^T P = I$ and $Q^T Q = I$, and $D$ is a diagonal matrix with positive $\sigma_1, \dots, \sigma_n$ on the diagonal. Then $A$ is invertible.
(E) Let $A = \begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{bmatrix} = \begin{bmatrix} A_{11} & X \\ 0 & Y \end{bmatrix}$, where $A_{11}, A_{12}, A_{21}, A_{22}, X$, and $Y$ are matrices, and $S = A_{22} - A_{21}A_{11}^{-1}A_{12}$ is called the Schur complement of $A_{11}$. Suppose that $A$ is invertible and $A_{11}$ is invertible, then $S$ is invertible.

Let $A = \begin{bmatrix} 1 & -6 & 4 \\ -6 & 2 & -2 \\ 4 & -2 & -3 \end{bmatrix}$. Giving an orthogonal matrix $P$ and a diagonal matrix $D$, we can orthogonally diagonalize $A$ in the way $A = PDP^{-1}$, where $P^{-1}$ is the inverse of $P$. Which of the following statements is/are true?
(A) The smallest eigenvalue of $A$ is -3.
(B) For the smallest eigenvalue of $A$, a basis for the eigenspace is $\begin{bmatrix} 2 \\ -2 \\ 1 \end{bmatrix}$. For the largest eigenvalue of $A$, a basis for the eigenspace is $\begin{bmatrix} -1 \\ -1 \\ 2 \end{bmatrix}$. For the third eigenvalue of $A$, a basis for the eigenspace is $\begin{bmatrix} 1 \\ 2 \\ 2 \end{bmatrix}$.
(C) $P$ can be constructed as $P = \frac{1}{3} \begin{bmatrix} 1 & -2 & 2 \\ 2 & -1 & -2 \\ 2 & 2 & 1 \end{bmatrix}$.
(D) $D$ can be constructed as $D = \begin{bmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{bmatrix}$, where $a, b$, and $c$ are the eigenvalues of $A$.
(E) $P^{-1} = P = P^T = \frac{1}{3} \begin{bmatrix} 1 & 2 & 2 \\ -2 & -1 & 2 \\ 2 & -2 & 1 \end{bmatrix}$.

Let the matrix $A = \begin{bmatrix} -3 & 2 \\ 6 & -4 \end{bmatrix}$. Let $\lambda$ be the eigenvalue of $A$, and $A^T$ be the transpose of $A$. Which of the following statements is/are true?
(A) The characteristic polynomial of $A^T A$ is $\lambda^2 - 117\lambda$.
(B) The singular values of $A$ are 117 and 0.
(C) Any factorization $A = U\Sigma V^T$, with $U$ and $V$ orthogonal, $\Sigma = \begin{bmatrix} D & 0 \\ 0 & 0 \end{bmatrix}$ and positive diagonal entries in $D$, is called a singular value decomposition (SVD) of $A$.
(D) Following (C), the matrices $U$ and $V$ are uniquely determined by $A$, and the diagonal entries of $\Sigma$ are the singular values of $A$.
(E) An SVD of $A$ is $\frac{1}{3}\begin{bmatrix} -1 & 2 \\ 2 & 1 \end{bmatrix} \begin{bmatrix} \sqrt{13} & 0 \\ 0 & \sqrt{13} \end{bmatrix} \frac{1}{\sqrt{5}}\begin{bmatrix} 3 & -2 \\ 2 & 1 \end{bmatrix}$.

Which of the following statements about the properties of matrices is/are true?
(A) An $m \times n$ matrix with more rows than columns has full rank if and only if its columns are linearly dependent.
(B) Suppose that $A$ is an $m \times n$ matrix such that for all $b$ in $\mathbb{R}^m$ the equation $Ax = b$ has at most one solution. Then the columns of $A$ must be linearly dependent.
(C) $\lambda$ is an eigenvalue of the matrix $A$ if and only if $\lambda$ is an eigenvalue of $A^T$, the transpose of $A$.
(D) If the matrices $A$ and $B$ are both orthogonally diagonalizable and $AB = BA$, then $AB$ is also orthogonally diagonalizable.
(E) The trace of a square matrix $A$, denoted by $\text{tr } A$, is the product of the diagonal entries in $A$. $\text{tr } (FG) = \text{tr } (GF)$ for any two $n \times n$ matrices $F$ and $G$.

Let $X$ be a continuous random variable that is uniformly distributed over $(0,1)$ and $Y = -\ln(X)$. Which of the following statements is/are true?
(A) $P(X > 0.25) = 0.75$.
(B) $E[X^2] = \frac{1}{3}$.
(C) $P(25X^2 - 20X + 3 > 0) > 0.55$.
(D) $E[Y] = 2$.
(E) $E[Y^2] = 2$.

Let $X$ be a Gaussian random variable such that $E[X] = 1$ and $E[(X-1)^2] = 4$. Let $Y$ be a Gaussian random variable such that $E[Y] = 0$ and $E[Y^2] = 1$. In addition, $X$ and $Y$ are statistically independent. Let $Z$ be a random variable such that $Z = 2X + 3Y$. Which of the following statements is/are true?
(A) $E[Z] = 2$.
(B) $E[Z^2] = 25$.
(C) $E[YZ] = 4$.
(D) $Y^2$ is an exponential random variable.
(E) $P(Z \ge 2) = \frac{1}{2}$.