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 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(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3.Consider the linear function L:R3 → R3. Suppose L(xi) = yi 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 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  (E) The intersection of R( A ) and N(AT) is the empty set.

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, 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 ATy = d cannot have a unique solution for any.

複選題7. Let Pn 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, .., Vm 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

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