111 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 and B be matrices in $\mathbb{R}^{n \times n}$. Which of the following statements are true?
(A) $\det(-A) = -\det(A)$.
(B) If $AA^T = I$, then $\det(A) = 1$.
(C) If $A A^T = I$, then $\text{trace}(A) = n$.
(D) If two rows of A are equal, then $\det(A) = 0$.
(E) If $\det(A) = \det(B)$, then A and B have the same rank.

Let $A \in \mathbb{R}^{3 \times 3}$ and its eigenvalues are $\lambda_1, \lambda_1$, and $\lambda_2$, where $\lambda_1$ and $\lambda_2$ are distinct eigenvalues. Suppose the dimension of $N(A - \lambda_1 I)$ is 1, where $N(A)$ denotes the null space of $A$. Which of the following statements are true?
(A) $\lambda_1$ must be a real number (not a complex number).
(B) $\lambda_2$ must be a real number (not a complex number).
(C) The dimension of $N(A - \lambda_2 I)$ equals 1.
(D) A is diagonalizable.
(E) A has two linearly independent eigenvectors corresponding to $\lambda_1$.

Let $A \in \mathbb{R}^{m \times n}$. Consider the linear equation $Ax = b$ or the homogeneous linear equation $Ax = 0$. Which of the following statements are true?
(A) If $\text{rank}(A) = m$, then $Ax = b$ has at least one solution for any $b \in \mathbb{R}^m$.
(B) If $\text{rank}(A) = m$, then $Ax = 0$ has only the trivial solution $x = 0$.
(C) If $\text{rank}(A) = n$, then $Ax = b$ has at most one solution for any $b \in \mathbb{R}^m$.
(D) If $\text{rank}(A) = n$ and $m > n$, then $Ax = 0$ has infinitely many solutions.
(E) If $\text{rank}(A) = m$ and $n > m$, then $Ax = 0$ has infinitely many solutions.

Let A and B be square matrices. Suppose that A is similar to B, that is, $B = P^{-1}AP$ for some nonsingular matrix P. Which of the following statements are true?
(A) If x is an eigenvector of B, then x is also an eigenvector of A.
(B) If y is in the column space of B, then y is also in the column space of A.
(C) $\text{trace}(A) = \text{trace}(B)$.
(D) A - I is similar to B - I.
(E) $A^5$ is similar to $B^5$.

Let $A \in \mathbb{R}^{m \times n}$, $R(A)$ 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 $y \in R(A)$, then $y \in R(AA^T)$.
(B) If $x \in N(A)$, then $x \in N(A^T A)$.
(C) $\text{rank}(A) + \dim(N(A)) = \text{rank}(A^T) + \dim(N(A^T))$.
(D) It is possible for a matrix A to have $[2, 1, -1]^T$ in $N(A)$ and $[1, 2, 3]^T$ in $R(A^T)$.
(E) Let $y \in \mathbb{R}^m$. If $y = u_1 + v_1 = u_2 + v_2$, where $u_1, u_2 \in R(A)$ and $v_1, v_2 \in N(A^T)$, then $u_1 = u_2$ and $v_1 = v_2$.

Let
$A = \begin{bmatrix} 4 & 1 & 3 & 2 \\ 1 & 4 & 3 & 3 \\ -1 & 1 & 1 & 7 \end{bmatrix}$.
Which of the following vectors are in the column space of A?
(A) $[3, 1, 2]^T$
(B) $[1, 0, -1]^T$
(C) $[0, 1, 3]^T$
(D) $[2, 1, 1]^T$
(E) $[4, 2, -1]^T$

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) $||u|| \geq ||p||$
(B) $||u - p|| \geq ||p||$
(C) $||u - p|| + ||p|| \geq ||u||$
(D) p = u implies $v = u$
(E) $|| p ||^2 + ||u - p||^2 = ||u||^2$

Suppose that vectors x and y in $\mathbb{R}^n$ satisfy $Rx = y$ for some $R \in \mathbb{R}^{n \times n}$ with $R^T R = I$. Which of the following options are possible?
(A) $x = [2, 1, 2]^T$, $y = [3, 0, 0]^T$.
(B) $x = [2, 1, 1]^T$, $y = [3, 1, 0]^T$.
(C) $x = [1, 2, 3]^T$, $y = [0, 0, 6]^T$.
(D) $x = [3, 4, 2]^T$, $y = [2, 0, 5]^T$.
(E) $x = [2, 3, 5]^T$, $y = [1, 5, 4]^T$.

Let L be the operator on $P_3$ (the set of polynomials of degree less than 3) defined by $L(p(x)) = xp'(x) + p''(x)$, where $p'(x)$ denotes the derivative of $p(x)$ with respect to x. Find the matrix A representing L with respect to $E = \{1 + x, -1 + x^2, 1 + x^2\}$. 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) $a_{21} = 0$
(B) $a_{22} = 0$
(C) $a_{31} = 0.5$
(D) $a_{32} = 1$
(E) $a_{33} = 2$

Let $S_{2 \times 2}$ be the set of all $2 \times 2$ symmetric matrices. Then $S_{2 \times 2}$ is a subspace of $\mathbb{R}^{2 \times 2}$. Suppose that $E = \{v_1, v_2, v_3\}$ and $F = \{w_1, w_2, w_3\}$ are two bases of $S_{2 \times 2}$, where
$v_1 = \begin{bmatrix} 1 & 3 \\ 3 & -7 \end{bmatrix}$, $v_2 = \begin{bmatrix} 1 & 1 \\ 1 & 17 \end{bmatrix}$, $v_3 = \begin{bmatrix} 0 & -3 \\ -3 & 7 \end{bmatrix}$
$w_1 = \begin{bmatrix} 4 & 1 \\ 1 & 0 \end{bmatrix}$, $w_2 = \begin{bmatrix} 1 & 1 \\ 1 & -3 \end{bmatrix}$, $w_3 = \begin{bmatrix} 0 & 3 \\ 3 & 1 \end{bmatrix}$
Let S be the transition matrix from E to F. Denote the first two rows of S by $[s_{11}, s_{12}, s_{13}]$ and $[s_{21}, s_{22}, s_{23}]$, respectively. Which of the following statements are true?
(A) $s_{11} = 1$
(B) $s_{12} = 1$
(C) $s_{13} = 3$
(D) $s_{21} = 3$
(E) $s_{22} = 1$

令 $F(s)=\mathcal{L}(f(t))$，$G(s)=\mathcal{L}(g(t))$。請問 $\mathcal{L}(3f(t)+g(t))$ 等於哪個函數？

令 $F(s)=\mathcal{L}(f(t))$，$G(s)=\mathcal{L}(g(t))$。請問 $\mathcal{L}(f(t)g(t))$ 等於哪個函數？

令 $F(s)=\mathcal{L}(f(t))$，且 $f(0)=0$。請問 $e^{3s}F(s)+s^2F(s)$ 是哪個函數的 Laplace 轉換？

令 $F(s)=\mathcal{L}(f(t))$。請問 $\mathcal{L}(t*f(t))$ 是哪個函數的 Laplace 轉換？

在這裡，$t*f(t)$ 代表函數 $g(t)=t$ 與函數 $f(t)$ 的 convolution。

令 $u(t)\equiv 0$。請問當參數 $a$ 與 $b$ 滿足甚麼條件的情況下，給予任何非零初值 $(x_1,x_2)$，方程式的解 $x(t)$ 都不會發散？

令 $u(t)\equiv 0$。請問當參數 $a$ 與 $b$ 滿足甚麼條件的情況下，給予任何非零初值 $(x_1,x_2)$，方程式的解 $x(t)$ 都會發散？

令 $u(t)\equiv 0$。請問當參數 $a$ 與 $b$ 滿足甚麼條件的情況下，給予任何非零初值 $(x_1,x_2)$，方程式的解 $x(t)$ 都會收斂到 $0$？

令 $u(t)\equiv 0$，$(a,b)=(0,-1)$。請列出所有會讓方程式的解收斂到 $0$ 的非零初值。