114 年 國立中山大學電波聯合碩士班《工程數學甲》

Consider the linear system $Ax = b$, where $A = [a_1, a_2, a_3] \in \mathbb{R}^{4 \times 3}$, $a_1, a_2, a_3$ are column vectors of A, and $b \in \mathbb{R}^4$ is a non-zero vector. Suppose
$a_1 + 2a_2 = 3a_3$
$a_1 + a_2 + a_3 = b$
$a_2 + 2a_3 \neq 0$

Which of the following statements are true?
(A) The linear system has a finite number of solutions.
(B) rank([A, b]) is equal to rank(A).
(C) The rank of A is less than or equal to 2.
(D) $x = [1, 2, 3]$ is a solution of the linear system.
(E) The vectors $a_1, a_2, a_3$ are linearly dependent.

Let

Which of the following statements are true?
(A) rank(A) = 2.
(B) trace(A) = 7.
(C) det(A) ≠ 0.
(D) 3 is an eigenvalue of A.
(E) All eigenvalues of A are non-negative.

Let

where $a_1, a_2, a_3$ are column vectors of $A \in \mathbb{R}^{3 \times 3}$, and $Q \in \mathbb{R}^{3 \times 3}$ is an orthogonal matrix, i.e., $Q^T Q = Q Q^T = I$. Which of the following statements are true?
(A) $a_1^T a_1 = 3$
(B) $a_1^T a_2 = 24$
(C) $a_1^T a_3 = 70$
(D) $a_2^T a_2 = 3$
(E) $a_2^T a_3 = 30$

Let

where $a_1, a_2, a_3$ are column vectors of A. Let $q_3$ be the orthogonal projection of $a_3$ onto $span(a_1, a_2)$, and $q_3 = [q_{31}, q_{32}, q_{33}, q_{34}]^T$. Which of the following statements are true?
(A) $q_{31} = 4$
(B) $q_{32} = 4/3$
(C) $q_{33} = 10/3$
(D) $q_{34} = 22/3$
(E) $a_3^T q_3 = 20$

Let $A \in \mathbb{R}^{n \times n}$, $A A^T = I$, and $A$ is a non-zero matrix. Which of the following statements are true?
(A) trace(A) ≠ 0.
(B) det(A) = 1.
(C) rank(A) = n.
(D) If n = 3, then trace(A) ≤ 3.
(E) All eigenvalues of A are real numbers.

Let $A \in \mathbb{R}^{m \times n}$, where $R(A)$ denotes the column space of A, $N(A)$ denotes the null space of A, and $\text{dim}(S)$ denotes the dimension of a subspace S. Which of the following statements are true?
(A) If A has linearly independent columns, then $A^T A$ is nonsingular.
(B) $\text{rank}(A^T) + \text{dim}(N(A^T)) = m$.
(C) $R(AA^T) = R(A^T A)$.
(D) If $AA^T$ is nonsingular, then $A^T A$ is also nonsingular.
(E) It is possible for a matrix A to have $[3, 4, 5]^T$ in $R(A)$ and $[1, -1, 2]^T$ in $N(A^T)$.

Find the inverse Laplace transform of $\frac{-5}{s+16}$.
(A) $5e^{16t}$
(B) $-5e^{16t}$
(C) $-5e^{-16t}$
(D) $5e^{-16t}$

Which of the following is wrong?
(A) The Fourier transform of a rectangular pulse of width T and height 1 is $T \text{sinc}(\frac{T\omega}{2})$.
(B) The Fourier transform of $u(t)$ (unit step function) is $\pi\delta(\omega)$.
(C) If $f(x)$ is odd, its Fourier series contains only sine terms.
(D) The initial value theorem for the Laplace transform is $\lim_{t \to 0^+} f(t) = \lim_{s \to \infty} sF(s)$.

Which of the following statements about the Gibbs phenomenon is wrong?
(A) It occurs near discontinuities when approximating a function using its Fourier series.
(B) The oscillatory behavior in the Gibbs phenomenon disappears completely as more terms are added to the Fourier series.
(C) The Gibbs phenomenon results in a characteristic overshoot near discontinuities, even with a large number of terms.
(D) The oscillations caused by the Gibbs phenomenon become narrower as more terms are included.

Suppose we are given the following information about a signal $x[n]$:
a. $x[n]$ is a real and even signal.
b. $x[n]$ has period $N=10$ and Fourier coefficients $a_k$.
c. $a_{11} = 5$.
d. $\frac{1}{10} \sum_{n=0}^{9} |x[n]|^2 = 50$.

Show that $x[n] = A \cos(Bn + C)$, and specify numerical values for the constants A, B, and C.

What is the Fourier transform of $\delta(t+1) + \delta(t-1)$?
(A) $e^{-j\omega} + e^{j\omega}$
(B) $e^{-j\omega} - e^{j\omega}$
(C) $2\cos(\omega)$
(D) $2\sin(\omega)$

The Fourier transform $X(e^{j\omega})$ of the discrete-time signal $x[n] = \cos(\frac{3\pi n}{5})$ is:
(A) $\pi [\delta(\omega - \frac{3\pi}{5}) + \delta(\omega + \frac{3\pi}{5})]$
(B) $\pi [\delta(\omega - \frac{3}{5}) - \delta(\omega + \frac{3}{5})]$
(C) $\sum_{k=-\infty}^{\infty} \pi [\delta(\omega - \frac{3\pi}{5} - 2\pi k) + \delta(\omega + \frac{3\pi}{5} - 2\pi k)]$
(D) $\sum_{k=-\infty}^{\infty} \pi [\delta(\omega - \frac{3\pi}{5} - 2\pi k) - \delta(\omega + \frac{3\pi}{5} - 2\pi k)]$

The solution to the differential equation $y'' - 2y' - 8y = f(t)$ with $y(0) = 1$ and $y'(0) = 0$, using the Laplace transform, is given by:
$y(t) = Ae^{4t} * f(t) + Be^{-2t} * f(t) + Ce^{4t} + De^{-2t}$.
where * denotes convolution. What are the correct values of A, B, C, and D?
(A) $A = \frac{1}{6}, B = -\frac{1}{6}, C = \frac{1}{3}, D = \frac{2}{3}$
(B) $A = \frac{1}{6}, B = \frac{1}{6}, C = -\frac{1}{3}, D = \frac{2}{3}$
(C) $A = \frac{1}{3}, B = -\frac{1}{3}, C = \frac{1}{6}, D = -\frac{2}{3}$
(D) $A = \frac{1}{6}, B = -\frac{1}{6}, C = -\frac{1}{3}, D = -\frac{2}{3}$

(7%) Magnetic resonance imaging is a medical imaging technique that uses macroscopic nuclear magnetization (M) of hydrogen in human tissues as its signal source. According to the well-known Bloch equation, the longitudinal magnetization Mz(t) follows the differential equation

where $M_0$ is the steady-state nuclear magnetization, and $T_1$ presents the longitudinal relaxation time, which is a tissue-dependent constant. Assume that an excitation pulse is applied to the magnetization right before $t = 0$ to remove $M_z$ completely; in other words, $M_z(0) = 0$. Find $M_z(t)$ on the interval $(0, \infty)$.

Use separation of variables to find the product solution of $u(x, y)$, which satisfies