110年 - 110 國立中山大學碩士暨碩士專班招生考試_部分碩士班:工程數學甲#105940

1.Let x and y be nonzero vectors in Rⁿ, n ≥ 2. Suppose x and y are not orthogonal, i.e. x^Ty ≠ 0. Let A = xy^T. Which of the following statements are true? (A) O is an eigenvalue of A. (B) x^Ty is an eigenvalue of A. (C) x is an eigenvector of A. (D) The number of linearly independent eigenvectors associated with eigenvalue O is n - 1. (E) A is diagonalizable.

3.Consider the linear function L: R³ → R³. Suppose L(x_i) = y_i fori = 1,2,3,4, where Suppose L(x₄) = [C₁, C₂,C₃]^T. Which of the following statements are true? (A)C₁=1 (B)c₁=2(C)c₂=2(D)c₂=3(E)c₃=-3

4.Let , R( A ) denote the column space of A, and N( A ) denote the null space of A. Which of the following statements are true? (A) There exists a nonzero vectorsuch that A^Ty = 0 and y = Ax for some (B) For every, we can always find some such that y = Ax. (C) Let p be the orthogonal projection of (D) Let y be a vector in R^m. If. (E) The intersection of R( A ) and N(A^T) is the empty set.

5.Let E ={V₁,V₂, V₃} be a basis for a three-dimensional subspace S of an inner product space V. Define. Suppose E is an orthogonal set with Let x = 5V₁ + 1V₂ -3V₃ and y = 1V₁- 3V₂ - 2V₃. Let p be the orthogonal projection of x onto y, and suppose p = cy, where o is a scalar. Which of the following statements are true? (A) (x,y) = 7 (B) (x,y) =8 (C) a = 7/33 (D)a = 8/14 (E) IIx||= 62

6.Let and nullity( A ) denote the dimension of N( A ). Suppose nullity( A ) = 2. Which of the following statements are true? (A) rank(A^T) = 3 (B) nullity(A^T) = 4 (C) Let A = [a₁, a₂, a₃], where a₁, ₂, a₃ are column vectors of A. Then a1 and az are linearly dependent. (D) The linear equation Ax = 0 has infinitely many solutions. (E) The lincar equation A^Ty = d cannot have a unique solution for any

7.Suppose that function f has period of 7, the Laplace transform of f is denoted by , andWhich of the following statements are true? (A)k₁+k₂=2(B)k₁-k₂=3(C)k₁+k₂=12(D)k₂+k₃=-10(E)k₂-k₃=5

8.Suppose that , the Laplace transform of f is denoted by  an Which of the following statements are true? (A)k₁=1 (B)k₂=9 (C)k₃ =3 (D)k₄ =9 (E) k₁ +k₂ + k₃ + k₄ =4

9. Suppose that f(t) satisfies the integral equation:

Solving the integral equation, we obtain Which of the following statements are (A)k₁+k₃=-1(B)k₂+k₃=1(C)k₁-k₃=2(D)k₂-k₃=3(E)k₂=-2

10. Suppose the Fourier transform of f is defined by, and
 where H(t) is the Heaviside function (unit step function). Which of the following statements are true? (A)k₁=11(B)k₂=-7 (C)k₃=3 (D)k₁ +k₂=-5 (E)k₂ +k₃=5

以下第11題到15題,考慮非線性微分方程式:统-以
11.令方程式的平衡點發生在 (A)(y1,y2)=(0,0) (B)(y₁,y₂)= (π,0)(C)以上皆是。

12.若μ＞0,則初值在平衡點(y₁,y₂)=(0,0)附近的解會隨時間增加而 (A)收斂到(0,0)(B)遠離(0,0)(C)不一定,跟初值有關。

13.若μ＜0,則初值在平衡點(y1,y2)=(0,0)附近的解會隨時間增加而 (A)收斂到(0,0)(B)遠離(0,0)(C)不一定,跟初值有關。

14.考慮對(y₁,y₂)=(π,0)作線性化之線性方程式。此時,若μ＜0,則該線性方程式之非零初值 解會隨時間增加而 (A)收斂到(0,0)(B)遠離(0,0)(C)不一定,跟初值有關。

15.考虑上题所述之線性方程式。此時,若μ＞0，則該線性方程式之非零初值解會隨時間增加而 (A)收斂到(0,0)(B)遠離(0,0)(C)不一定,跟初值有關。

16.考慮複函數f(z)=cosz。 (A)|f(z)|隨z的虛部增大而發散(B)|f(z)|隨z的實部增大而發散(C)。

17.考慮複函數。 (A)原點是f(z)的唯一一個極點(pole) (B)f(z)在原點之外的所有複平面上皆為解析(analytic)。 (C)以上皆對。

(a).(10分)令,請找出微分方程式之解。

(b).(5分)請問(a)小題的解是否滿足?

(c)(5分)令。請問此時微分方程式的解是否滿足。

(a).(5分),其中C為沿著頂點為±2與±2i之正方形四邊正向旋轉一周之封閉 路徑。

(b).(10分)正向旋轉一周之封閉路徑。