114 年 國立中山大學電機工程學系碩士班庚組《工程數學甲》

Consider the linear system Ax = b, where A = [a1, a2, a3] ∈ R4×3, a1, a2, a3 are column vectors of A, and b ∈ R4 is a non-zero vector. Suppose
a1 + 2a2 = 3a3, a1 + a2 + a3 = b, a2 + 2a3 ≠ 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 a1, a2, a3 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 ∈ R3×3, and Q ∈ R3×3 is an orthogonal matrix, i.e., $Q^T Q = QQ^T = I$ is the identity matrix. Which of the following statements are true?
(A) $a_1^T a_1 = 3$
(B) $a_2^T a_2 = 24$
(C) $a_3^T a_3 = 70$
(D) $a_1^T a_1 = 3$
(E) $a_1^T a_2 = 30$

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 $A \in R^{n \times n}$, $A A^T = I$, 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 ∈ R^m×n, where R(A) denotes the column space of A, N(A) denotes the null space of A, and 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^TA is nonsingular.
(B) rank(A^T) + dim(N(A^T)) = m.
(C) R(AA^T) = R(A^TA).
(D) If AA^T is nonsingular, then A^TA 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 $\frac{-5}{s^2 + 16}$.
(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 $T \text{sinc}(\frac{\omega T}{2})$.
(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. $\sum_{n=0}^{9} |x[n]|^2 = 50$.
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(ejω) of the discrete-time signal x[n] = cos($\frac{3\pi n}{5}$) is:
(A) $\frac{\pi}{5}(\delta(\omega - \frac{3\pi}{5}) + \delta(\omega + \frac{3\pi}{5}))$
(B) $\pi\delta(\omega - \frac{3\pi}{5}) - \pi\delta(\omega + \frac{3\pi}{5})$
(C) $\sum_{l=-\infty}^{\infty} \pi [\delta(\omega - \frac{3\pi}{5} - 2\pi l) + \delta(\omega + \frac{3\pi}{5} - 2\pi l)]$
(D) $\sum_{l=-\infty}^{\infty} \pi [\delta(\omega - \frac{3\pi}{5} - 2\pi l) - \delta(\omega + \frac{3\pi}{5} - 2\pi l)]$

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) $A = \frac{1}{6}, B = -\frac{1}{3}, C = \frac{1}{6}, D = -\frac{2}{3}$
(B) $A = \frac{1}{6}, B = -\frac{1}{3}, C = -\frac{1}{3}, D = \frac{2}{3}$
(C) $A = -\frac{1}{6}, B = \frac{1}{3}, C = \frac{1}{3}, D = -\frac{2}{3}$
(D) $A = \frac{1}{6}, B = \frac{1}{3}, C = -\frac{1}{3}, D = -\frac{2}{3}$

(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 M₂(t) follows the differential equation
$\frac{dM_z(t)}{dt} = \frac{M_0 - M_z(t)}{T_1}$
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 M₂ completely; in other words, $M_z(0) = 0$. Find $M_z(t)$ on the interval $(0,\infty)$.

(8%) Use separation of variables to find the product solution of u(x, y), which satisfies
$\frac{1}{y} \frac{\partial u}{\partial x} = \frac{1}{3x} \frac{\partial u}{\partial y}$

(10%) Given that an 3D object is enclosed by a surface $f(x, y, z) = 0$ and the xy plane (i.e., $z = 0$), where $f(x, y, z) = x^2 + y^2 + z - 2$.
(1) (4%) Find the normal vector over the surface of this object. Note that the normal vector should be provided in the form of a unit vector here.
(2) (6%) What is the surface area of this object?

(10%) Given that z is a complex variable and $\bar{z}$ represents its complex conjugate.
(1) (2%) Sketch the set of points determined by $z\bar{z} = 2$ on the complex plane.
(2) (8%) Let C be the contour of your answer in (1) in a positive sense. Solve
$\oint_C \frac{e^{-z}}{z^2 + 1} dz$