複選題

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

複選題1.Let x and y be nonzero vectors in Rn, n ≥ 2. Suppose x and y are not orthogonal, i.e. xTy ≠ 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 O is n - 1. (E) A is diagonalizable.

複選題2.Let A and B be square matrices and a be a real number. Which of the following statements are true? (A)det(A2) = det(ATA) (B) det(BTA) = det(ATBT) (C) det(aA) = aㆍdet( A ) (D) det(A+B) = det( A ) + det ( B ) (E) Swapping two rows of A will not change its determinant.

複選題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 ATy = 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 Rm. If. (E) The intersection of R( A ) and N(AT) is the empty set.

複選題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

複選題6.Let and nullity( A ) denote the dimension of N( A ). Suppose nullity( A ) = 2. Which of the following statements are true? (A) rank(AT) = 3 (B) nullity(AT) = 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 ATy = d cannot have a unique solution for any

複選題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

複選題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

複選題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

複選題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。

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