110年 - [非官方正解]110 國立中山大學_碩士班招生考試_電機系(甲、戊、己、庚組)、通訊所(乙組)、電波聯合：工程數學甲#105948

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 = xyT. Which of the following statements are true? (A) O is an eigenvalue of A. (B) xTy is an eigenvalue of A. (C) x is an eigenvector of A. (D) The number of linearly independent eigenvectors associated with eigenvalue 0 is n - 1. (E) A is diagonalizable.

3.Consider the linear function L:R³ → R³. Suppose L(x_i) = y_i for i = 1,2,3,4, whereSuppose 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 vector such 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  (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 Let p be the orthogonal projection of x onto y, and suppose p = ay, where a 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) lIxll = 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, a₂, a₃ are column vectors of A. Then a₁ and a₂ are linearly dependent. (D) The linear equation Ax = 0 has infinitely many solutions. (E) The linear equation A^Ty = d cannot have a unique solution for any.

8. Let and Let p be the orthogonal projection of b onto N( A ). Suppose b - p = av₁ + βv₂. Which of the following statements are true? (A)9a=8 (B) 9a= 9 (C) 9 β = 7 (D) 9 β=8 (E) 9a + 9 β = 15

9.Define a mapping L: R³ →R³ by L(x) = Kx, where . Suppose that E = {p, Kp, K²p} is an ordered basis of R³, and K³p - 5K²p + 7Kp - 11p = 0. Let A be the matrix representation of L with respect to basis E. SupposeWhich of the following statements are true? (A) (B)(C) (D)(E)

10. Let S be the transition matrix from {V₁,V₂} to {u₁, u2}. Suppose Which of the following statements are true? (A)a=-1 (B)b= 2(C)c=3 (D)d=4 (E) a +b+c+d= 23

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。

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.考慮上題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。

 下面16-20题為單選題,每题1分,總分5分。答錯或未作答者該題以0分計 考慮函數(R代表實数所成之集合)及Laplace轉换,定義為: 以下第16题到20题,我們令F(s)=L(f(t))。
16.請問Laplace轉换是否為一個線性操作? (A)是(B)否(C)不一定,與函數f有關。

17.等於以下哪個函數? (A)(B) sF(s)(C)以上兩者都不是。

18.令θ>0為一實數,則L(f(t-θ))等於以下哪個函数? (A) F(s-θ) (B)(C) F(θs)

19.L(tf(t))等於以下哪個函數? (A) (B)sF(s) (C)

20.令n為一正整數,f(t)=tⁿ,則L(f(t))等於以下哪個函數? (A) (B) (C)