112 年 國立臺灣大學電機工程研究所甲組《工程數學（含線性代數、微分方程）》

1. (7%) Determine which of the following statements are true.
   (A) The solution set of the matrix equation $X^2 + 3X + 2I_2 = 0$ with the $2 \times 2$ matrix variable $X$ is equal to $\{-I_2, -2I_2\}$.
   (B) If the $n \times n$ matrix $M$ is skew-symmetric, i.e., $M = -M^T$, then $M$ is not invertible when $n$ is an odd positive integer.
   (C) Let $T_A$ be the matrix transformation induced by the matrix $A$. For every square matrix $A$, Range($T_A$) $\cap$ Null($T_A$) = $\{0\}$.
   (D) If $A$ is an $m \times n$ matrix and $E$ is an $n \times n$ elementary matrix, then Null $A$ = Null $AE$.
   (E) None of the above statements are true.

2. (7%) Which of the following subsets of $\mathbb{R}^n$ are a subspace.
   (A) $S_1 = \{ \begin{pmatrix} s \\ t \\ u \end{pmatrix} | s, t, u \in \mathbb{R}, s + t + u = 1 \}$
   (B) $S_2 = \{ \begin{pmatrix} x \\ y \\ z \end{pmatrix} | x, y, z \in \mathbb{R}, xyz = 0 \}$
   (C) $S_3 = \{ v \in \mathbb{R}^4 | \begin{pmatrix} 1 & 2 & 3 & 4 \\ 8 & 7 & 6 & 5 \\ 1 & 1 & 1 & 0 \end{pmatrix} v = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix} \}$
   (D) $S_4 = \text{Span} \left\{ \begin{pmatrix} 1 \\ 2 \\ 4 \end{pmatrix}, \begin{pmatrix} 2 \\ 3 \\ 6 \end{pmatrix}, \begin{pmatrix} 3 \\ 6 \\ 9 \end{pmatrix} \right\}$
   (E) $S_5 = \text{Span} \left\{ \begin{pmatrix} 1 \\ 1 \\ 3 \end{pmatrix}, \begin{pmatrix} 4 \\ 7 \\ 9 \end{pmatrix} \right\} \cup \text{Span} \left\{ \begin{pmatrix} 1 \\ 1 \\ 2 \end{pmatrix}, \begin{pmatrix} 3 \\ 6 \\ 1 \end{pmatrix} \right\}$

3. (7%)
   $A = \begin{pmatrix} 1 & 2 & 0 \\ 1 & 1 & 1 \\ 0 & 1 & 1 \end{pmatrix}$, $S = \left\{ \begin{pmatrix} -3 \\ 0 \\ 1 \end{pmatrix}, \begin{pmatrix} -1 \\ 1 \\ 2 \end{pmatrix}, \begin{pmatrix} -1 \\ -1 \\ 4 \end{pmatrix} \right\}$. Which of the following statements are true?
   (A) Let $T_A$ be the matrix transformation induced by the matrix $A$. $T_A$ is one-to-one.
   (B) Let $T_A$ be the matrix transformation induced by the matrix $A$. $T_A$ is onto.
   (C) The dimension of the range of $T_A$ is equal to 2.
   (D) Null $A = \text{Span } S$.
   (E) Let $B_1$ be any basis for the range of $T_A$ and $B_2$ be any basis for Span $S$. Then $B_1 \cup B_2$ constitutes a basis for $\mathbb{R}^4$.

4. (7%) Let $A$ and $B$ be $m \times n$ and $n \times k$ matrices respectively. Which of the following statements are true?
   (A) Col $A$ is a subset of Col $AB$.
   (B) $\dim(\text{Col } A) \le \dim(\text{Col } AB)$.
   (C) $\text{rank } A \ge \text{rank } AB$.
   (D) Suppose that $k = n$ and $B$ is an $n \times n$ invertible square matrix. Then Col $AB$ is a subset of Col $A$.
   (E) None of the above statements are true.

5. (7%) Let $Q$ be an $m \times n$ invertible matrix and $P$ be an $n \times n$ matrix. Let $R_1$ and $R_2$ be the reduced row echelon form of $Q$ and $P$ respectively. Which of the following statements are true?
   (A) $R_1$ is an identity matrix.
   (B) If $R_3$ is also the reduced row echelon form of $P$, then $R_3$ must be equal to $R_2$.
   (C) $\det(Q) = \det(R_1)$.
   (D) Let $\{u_1, u_2, \dots, u_k\}$ be a linearly independent set of vectors in $\mathbb{R}^n$. Then $\{Pu_1, Pu_2, \dots, Pu_k\}$ is linearly independent.
   (E) Suppose that $\{Pu_1, Pu_2, \dots, Pu_k\}$ is linearly independent. It is necessary that $k \le n$.

6. (7%) Determine which of the following statements are true?
   (A) Let $B$ be a basis of $\mathbb{R}^m$ and $A$ be an $n \times n$ matrix. If $||Av|| = ||v||$ for every vector $v \in B$, then $A$ is orthogonal.
   (B) If two square matrices have the same characteristic polynomials, then they are similar.
   (C) In an inner-product space $V$, if $\langle u, v \rangle = \langle w, v \rangle$ for every vector $v$ in a basis for $V$, then $u = w$.
   (D) If two vectors are linearly independent and they are both eigenvectors of a symmetric matrix, then they are orthogonal.
   (E) Let $A$ be an $m \times n$ matrix and $W = \text{Col } A$. If $P_W$ is the orthogonal projection matrix for $W$, then $Ax = P_W b$ is consistent for every $b \in \mathbb{R}^m$.

7. (8%) Let $T: P_2 \to P_2$ be defined by
   $T(p(x)) = 3p(0) + (-2p(1) + p'(0))x + p(2)x^2$,
   where $p'(0) = \frac{dp(x)}{dx}\bigg|_{x=0}$.
   (A) $T$ is a linear transformation.
   (B) $T$ is an isomorphism.
   (C) The eigenvalues of $T$ are 0 and -3.
   (D) A set of vectors consisting of the basis of each eigenspace for all eigenvalues of $T$ constitutes a basis for $\mathbb{R}^3$.
   (E) Let $\{1, x, x^2\}$ be a basis for $P_2$. Then any vector $v$ in vector space $P_2$ can be uniquely represented as a linear combination of the vectors in $\{1, 1+2x, 1+2x+3x^2\}$.

For the following problems (8 to 17), assume: $y' = \frac{dy}{dx}$ and $y'' = \frac{d^2y}{dx^2}$

8. (5%) The following differential equation can be classified as:
   $xy' + xy + 2y = x^2e^{-x}$
   (A) linear homogenous
   (B) linear nonhomogenous
   (C) nonlinear homogenous
   (D) nonlinear nonhomogenous
   (E) cannot determine

9. (5%) For the following differential equation, $x = -3$ is which type of point?
   $(x^2+x-6)y'' + (x+3)y' + (x-2)y = 0$
   (A) ordinary
   (B) regular singular
   (C) irregular singular
   (D) none of these

10. (5%) Which is a solution of the following differential equation on interval $(-5, 5)$?
    $\frac{dy}{dx} = \frac{x}{y}$
    (A) $x^2 + y^2 = 25, -5 < x < 5$
    (B) $y = \sqrt{25-x^2}, -5 < x < 5$
    (C) $y = -\sqrt{25-x^2}, -5 < x < 5$
    (D) all of the above
    (E) none of these

11. (5%) Which is a solution to the following differential equation?
    $y' + y = 0$
    (A) $y = 0$
    (B) $y = e^x$
    (C) $y = \sum_{n=0}^\infty \frac{(-1)^n}{n!} x^n$
    (D) both A and B
    (E) all of the above

12. (5%) Which is the general solution to the following differential equation?
    $y''' - 3y'' + 3y' - y = 0$
    Assume $c_1, c_2$, and $c_3$ are arbitrary constants.
    (A) $y = c_1 e^x + c_2 e^x + c_3 e^x$
    (B) $y = c_1 e^x + c_2 e^{x+1} + c_3 e^{x+2}$
    (C) $y = c_1 e^x + c_2 x e^x + c_3 x^2 e^x$
    (D) $y = c_1 e^x + c_2 \ln x e^x + c_3 (\ln x)^2 e^x$
    (E) none of these

13. (5%) Which is the general solution to the following differential equation?
    $x^2y'' - 2xy' - 4y = 0$
    Assume $c_1$ and $c_2$ are arbitrary constants.
    (A) $y = e^x [c_1 \cos(\sqrt{5}x) + c_2 \sin(\sqrt{5}x)]$
    (B) $y = c_1 e^x + c_2 e^{4x}$
    (C) $y = x [c_1 \cos(\sqrt{5} \ln x) + c_2 \sin(\sqrt{5} \ln x)]$
    (D) $y = c_1 x^{-1} + c_2 x^4$
    (E) none of these