110年 - 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

選擇題 (20)

複選題

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.

複選題
2.Let A and B be square matrices and o be a real number. Which of the following statements are true? (A) det(A²) = det(A^TA) (B) det(B^TA) = det(A^TB^T) (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

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

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 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.

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 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

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(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.

複選題
7. Let P_n denote the set of all polynomials of degree less than n. Which of the following statements are true? (A) Suppose V₁, V₂, V₃ are linearly independent vectors in P₃. Then V₁, V₂, V₃ span P₃. (B) If V₁,V₂, V₃ are linearly dependent, then Span(v₁, v₂) must be equal to Span(v₂, v₃). (C) P₂ is a subspace of P₃. (D) Ifv₁ + v₂ + V₃ equals the zero polynomial, then V₁, V₂, V₃ must be linearly dependent. (E) If V₁,V₂, .., V_m span a vector space V, then V₁, V₂, ..., Vm must be linearly independent.

複選題

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

複選題

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)

複選題

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