114 年 國立中山大學電機工程學系碩士班乙組《工程數學乙》

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) $x = [1,2,3]^T$ 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($A$) $\neq$ 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., $Q^T Q = QQ^T = I$ 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 $A \in \mathbb{R}^{n \times n}$, $A A^T = I$, and $A$ is a non-zero matrix. Which of the following statements are true?
(A) trace($A$) $\neq$ 0.
(B) det($A$) = 1.
(C) rank($A$) = n.
(D) If n = 3, then trace($A$) $\le$ 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 $dim(S)$ denotes the dimension of a subspace $S$. Which of the following statements are true?
(A) If $A$ has linearly independent columns, then $A A^T$ is nonsingular.
(B) rank($A$) + dim($N(A^T)$) = m.
(C) $R(A A^T) = R(A^T A)$.
(D) If $A A^T$ 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 $N(A^T)$.

Let $A \in \mathbb{R}^{3 \times 3}$ with rank($A^2$) = rank($A$) - 1, and $A$ has at least two distinct eigenvalues. Which of the following statements are true?
(A) $A$ is diagonalizable.
(B) $A$ is not symmetric.
(C) rank($A^2$) = 1.
(D) trace($A$) $\neq$ 0.
(E) $A$ is a nonsingular matrix.

Let $u$ and $v$ be two vectors in an inner product space $V$, and $p$ is the orthogonal projection of $u$ onto $v$. Define $||v|| = \sqrt{(v, v)}$. Which of the following statements are true?
(A) $||u|| \ge ||u - p||$.
(B) $||p|| \ge ||u - p||$.
(C) $p = u$ if and only if $u$ and $v$ are orthogonal.
(D) $||u - p||^2 = ||u||^2 - ||p||^2$.
(E) $||u - p|| \le ||u - \alpha v||$ for any $\alpha \in \mathbb{R}$.

Let $\mathbb{P}^2$ be the space of all real-valued polynomials of degree at most $2$, with the inner product defined as

where $f(x),g(x)\in\mathbb{P}^2$. Define $\|f\|=\sqrt{\langle f,f\rangle}$. Consider the orthogonal set of polynomials $\{u_1,u_2,u_3\}$ in $\mathbb{P}^2$ with respect to this inner product, where

Suppose $p(x)\in\mathbb{P}^2$ is a polynomial such that $\langle p,u_1\rangle=6$, $\langle p,u_2\rangle=-6$, and $\langle p,u_3\rangle=4$. Let the orthogonal projection of $p(x)$ onto $\operatorname{span}(u_1,u_2)$ be denoted by $q(x)$. Which of the following statements are true?

(A) $\langle u_3,u_3\rangle=\frac{2}{3}$.
(B) $q(x)=2u_1(x)-3u_2(x)$.
(C) $\|q(x)\|^2=30$.
(D) $p(x)=6u_1(x)-6u_2(x)+4u_3(x)$.
(E) $\|p(x)\|^2=\frac{572}{3}$.

Let $L:\mathbb{R}^3\to\mathbb{R}^3$ be a linear operator defined as

where $d_1,d_2,d_3$ are given scalars. Let $A$ be the matrix representation of $L$ with respect to the standard basis $E=\{\mathbf{e}_1,\mathbf{e}_2,\mathbf{e}_3\}$ of $\mathbb{R}^3$, where $\mathbf{e}_1=[1,0,0]^T$, $\mathbf{e}_2=[0,1,0]^T$, and $\mathbf{e}_3=[0,0,1]^T$. Denote the last two rows of $A$ by $[a_{21},a_{22},a_{23}]$ and $[a_{31},a_{32},a_{33}]$, respectively. Which of the following statements are true?

(A) $a_{21}=d_2$.
(B) $a_{22}=d_1$.
(C) $a_{31}=-d_3$.
(D) $a_{32}=d_1$.
(E) $a_{33}=0$.

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 $M_z(t)$ follows the differential equation

where $M_0$ is the steady-state nuclear magnetization, and $T_1$ represents 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 $(0,\infty)$.

Find the general solution of $y(x)$ for

Find the general solution of $x(t)$ and $y(t)$ satisfying the following equations:

(1) (12%) Find the power series solution of

about the point $x=0$.

(2) (2%) What is the radius of convergence for your solution in (1)?