115 年 國立中山大學電機工程學系碩士班乙組《工程數學乙》

Let $a_1, a_2, a_3, a_4, d \in \mathbb{R}^n$. Define the matrices
$B = [a_1, a_2, a_3] \in \mathbb{R}^{n \times 3}$, $C = [a_1, a_2, a_3, a_4] \in \mathbb{R}^{n \times 4}$.
Assume that the linear system $Bx = d$ has no solution for $x \in \mathbb{R}^3$, and the linear system $Cy = d$ has at least one solution for $y \in \mathbb{R}^4$. Let $[B, d] = [a_1, a_2, a_3, d] \in \mathbb{R}^{n \times 4}$. Which of the following statements are true? (Select all that apply.)
(A) The set $\{a_4, d\}$ is linearly dependent.
(B) The set $\{a_1, a_2, a_3, d\}$ is linearly dependent.
(C) The set $\{a_1, a_2, a_3, a_4\}$ is linearly independent.
(D) $\text{rank}(C) > \text{rank}(B)$.
(E) $\text{rank}(C) = \text{rank}([B, d])$.

Let $A \in \mathbb{R}^{m \times n}$. For any matrix $M$, let $R(M)$ denote the column space and $N(M)$ denote the null space. Which of the following statements are true? (Select all that apply.)
(A) If $y \in R(A A^T)$ and $y \neq 0$, then $A y \neq 0$.
(B) If $y \in R(A)$, then $A y = 0$.
(C) If $m = n$, then $R(A) = R(A^T)$.
(D) If $m = n$, then $N(A) = N(A^T)$.
(E) If $x \in \mathbb{R}^n$ and $A x \neq 0$, then $x \in R(A^T)$.

Let $A \in \mathbb{R}^{m \times n}$, $B \in \mathbb{R}^{m \times k}$, and $C \in \mathbb{R}^{k \times n}$ such that $A = BC$. For any matrix $M$, let $R(M)$ denote the column space and $N(M)$ denote the null space. Which of the following statements are true? (Select all that apply.)
(A) $R(B)$ is a subset of $R(A)$.
(B) $N(A)$ is a subset of $N(C)$.
(C) $k \ge \text{rank}(A)$.
(D) If $k > \text{rank}(A)$, then $\text{rank}(B) = \text{rank}(C)$.
(E) If $k = \text{rank}(A)$, then $\text{rank}(A) = \text{rank}(B)$.

Let
$A = \begin{pmatrix} 3 & 0 & 3 & 0 & 3 \\ 0 & 2 & 0 & 2 & 0 \\ 3 & 0 & 3 & 0 & 3 \\ 0 & 2 & 0 & 2 & 0 \\ 3 & 0 & 3 & 0 & 3 \end{pmatrix}$
Which of the following statements are true? (Select all that apply.)
(A) $A$ is diagonalizable (over $\mathbb{R}$).
(B) $\text{rank}(A) = 2$.
(C) $2$ is an eigenvalue of $A$.
(D) $3$ is an eigenvalue of $A$.
(E) The number of positive eigenvalues of $A$ is 2.

Let
$v_1 = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, \quad v_2 = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}, \quad x = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}$
and define the subspace $W = \text{span}\{v_1, v_2\} \subset \mathbb{R}^3$. Let $p$ be the orthogonal projection of $x$ onto $W$. For any vector $u$, let $||u||$ denote the Euclidean norm: $||u|| = \sqrt{u^T u}$. Which of the following statements are true? (Select all that apply.)
(A) $||p|| = 2$.
(B) $||x - p|| = 3$.
(C) $w^T (x - p) = 0$ for every $w \in W$.
(D) $x - p \in W$.
(E) $p$ is orthogonal to $v_1$.

Let
$b_1 = \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}, \quad b_2 = \begin{pmatrix} -1 \\ 1 \\ 0 \end{pmatrix}, \quad b_3 = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}, \quad x = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}$
Define a linear transformation $L: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ satisfying
$L(b_1) = 2b_1$, $L(b_2) = b_2$, $L(b_3) = -b_3$.
Let $L^5(x) = L(L(L(L(L(x)))))$. Suppose that, in standard coordinates (with respect to the standard basis of $\mathbb{R}^3$), $L^5(x) = [y_1, y_2, y_3]^T \in \mathbb{R}^3$. Let $A$ be the matrix representation of $L$ with respect to the ordered basis $\{b_1, b_2, b_3\}$. Which of the following statements are true? (Select all that apply.)
(A) $y_1 = 33$.
(B) $y_2 = 31$.
(C) $y_3 = -3$.
(D) $\det(A) = -2$.
(E) $\text{rank}(A) = 2$.

Let $A \in \mathbb{R}^{4 \times 4}$. Suppose the following ranks are given:
$\text{rank}(A - I) = 3$, $\text{rank}(A - 2I) = 2$, $\text{rank}(A^2 - I) = 2$.
Which of the following statements are true? (Select all that apply.)
(A) $A$ is symmetric.
(B) $A$ is not diagonalizable (over $\mathbb{R}$).
(C) $\det(A) = 4$.
(D) $\text{trace}(A) = 4$.
(E) $\text{rank}(A) = 4$.

Let $K\in\mathbb{R}^{3\times 3}$ and define a linear mapping $L:\mathbb{R}^{1\times 3}\to\mathbb{R}^{1\times 3}$ by $L(x)=xK$. Let $c$ be such that the ordered set $E=\{c,cK,cK^2\}$ is a basis of $\mathbb{R}^{1\times 3}$. Suppose $c$ satisfies

Let $A$ be the matrix representation of $L$ with respect to $E$. $A$ is defined by $[L(x)]_E=A[x]_E$, where $[x]_E\in\mathbb{R}^3$ is the column coordinate vector of $x$ in the basis $E$. 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? (Select all that apply.)

(A) $a_{22}=0$
(B) $a_{23}=1$
(C) $a_{31}=0$
(D) $a_{32}=-7$
(E) $a_{33}=5$

Let $u,v\in\mathbb{R}^n$ be nonzero vectors with $n\geq 2$ and $u^Tv\neq 0$. Define

Which of the following statements are true? (Select all that apply.)

(A) $v$ is an eigenvector of $M$.
(B) $1$ is an eigenvalue of $M$.
(C) $M$ is invertible and $M^{-1}=M$.
(D) Every eigenvalue of $M$ is real.
(E) $M$ is symmetric.

Let $V=\mathbb{R}^{2\times 2}$ with inner product $\langle A,B\rangle=\operatorname{trace}(A^TB)$ and define $\|A\|=\sqrt{\langle A,A\rangle}$. Consider the following set $\{U_1,U_2,U_3,U_4\}\subset V$, which is orthogonal with respect to the given inner product:

Suppose $M\in V$ satisfies

Let $P$ be the orthogonal projection of $M$ onto $\operatorname{span}\{U_1,U_2\}$. Which of the following statements are true? (Select all that apply.)

(A) $P=2U_1-U_2$
(B) $\|P\|^2=20$
(C) $M=2U_1+U_2-U_4$
(D) $\|M\|^2=20$
(E) $\langle M,P\rangle=24$

Given the differential equation (DE)

where $x$ is the independent variable. Which of the following is/are INCORRECT?

(A) This DE is nonlinear.
(B) This DE is nonhomogeneous.
(C) The solution of this DE is $y=\frac{\ln(x+1)}{x+1}+\frac{c}{x+1}$.
(D) The solution interval is $(-1,\infty)$.
(E) Provided the initial condition $y(e)=1$, there exists a unique solution.

Given the differential equation (DE)

where $x$ is the independent variable. Which of the following is/are INCORRECT?

(A) This DE is exact.
(B) Consider $M(x,y)=x^2+2y^2$ and $N(x,y)=-(xy)$. $M$ and $N$ are homogeneous functions.
(C) One of the solutions of this DE is $x^4=(x^2+y^2)/2$, which is an explicit solution.
(D) Consider the initial condition $y(-1)=1$. The solution interval is $(-\infty,\infty)$.
(E) There exists a singular point.

Given

which of the following is correct? Assume $t$ is the independent variable.

(A) There are two distinct eigenvalues.
(B) The stationary point is a repeller.
(C) All solutions will converge onto the origin.
(D) One of the eigenvalues is $\lambda_2=1$.
(E) One of the solution vectors is

Consider the differential equation (DE)

Which of the following is/are correct?

(A) There is an attractor.
(B) $y(x)=2$ is a singular solution.
(C) $y(x)=\frac{ce^x-2}{ce^x-1}$ is a one-parameter family of solutions, where $c$ is an arbitrary constant.
(D) Provided $y(0)=-1$, the largest interval is $[0,\infty)$.
(E) Given $y(0)=4$, there exists an interval centered at $0$ on which the DE has a unique solution.

Here is the differential equation $y'=y^2e^{-x}$, with $x$ being the independent variable. Which of the following is/are correct?

(A) This DE is exact.
(B) There is no singular solution.
(C) The solutions $y(x)>0$ or $y(x)<0$.
(D) The solution of this DE is $y=\frac{1}{e^{-x}+c}$.
(E) Given the initial condition $y(0)=1$, the solution interval is $(-\infty,\infty)$.

Which of the following is/are INCORRECT?

(A) Consider $xy''+(\sin x)y=0$. $x=0$ is a regular singular point.
(B) Consider $x^3y''-xy'+y=0$. $x=0$ is a regular singular point.
(C) Consider $x^3y''-xy'+y=0$. There exists one solution that is analytic at $x=0$.
(D) Consider $y''+\ln(x+1)y'+y=0$. $x=0$ is a regular singular point.
(E) Consider $y''+\ln(x+1)y'+y=0$. The solutions centered at $x=2$ must converge between $[-1,1]$.

Given the differential equation (DE)

where $x$ is the independent variable. On the way to finding infinite series solutions around $x=0$, which of the following are correct?

(A) There is no trivial solution.
(B) One of the indicial roots is $0$.
(C) One of the solutions is a power series.
(D) The solution interval is $(0,\infty)$.
(E) Given the form of a solution $y=\sum\frac{A^n}{B^n n!}x^n$, where $n$ is a nonnegative integer, $A+B=1$.