10. Let P ∈ be the orthogonal projection matrix, with respect to the Euclidean inner
product, of vectors in onto col( A ) for some nonzero matrix A ∈ . Which of the
following statements is/are true?
(A) ≥rank(P).
(B) P can have an eigenvalue larger than 2.
(C) Px≠ x for all x e .
(D) If k ＜ n and ! e , the system of linear equations Ax = o has a unique least squares
solution.

(E) None of the above is true.

6. For the matrixwhich of the following statements is/are true?(A) rank( A ) = 3.(B) The sum of eigenvalues of A is 6.(C) A is similar to B =(D) The system of linear equations A x = has a solution.(E) None of the above is true.

複選題7. For a non-zero matrix A ∈ , deine(1)the maximum Frobenius norm ||A  x llover all unit-norm vectors  x . Which of the followingstatements is/are true?(A) σ2 is the maximal eigenvalue of .(B) If x0 is an optimal solution to equation (1), i.e, llAx0ll = σ, then there exists a vectorx1 such that Ax1 = 0  and ≠0.(C) Assune m = n and A is nonsingular. Then = .(D) Assume m = n. Then t A is singular only if σ ≥ 1.(E) None of the above is true.

複選題8. Let V together with the inner product (‧,‧)y be an inner product space over . For acollection of N linearly independent vectors [w1 ,...,] in V, let M = bethe corresponding Gram matrix, i.e, the (i, j)-th entry of M is given by Define the following set of real-valued functions f : V →which is a vector space over when combined with standard addition and scalar multiplica-M= 1.8m (2m,3)v tion of real-valued functions. For any f( x )  = in the vector space F, defineWhich of the following statements is/are true?(A) M is positive definite.(B) If m1,1 = 2, m2,2 = 1, and m1,2 = -, then is the vector norm induced by the inner product .(C) (f,g)s is an inner product for elements f,g ∈ F.(D) Given g(y) = , we have = f(ω1) for all f ∈ F.(E) None of the above is true.＜f(z) =

9. Which of the following staternents is/are true?(A) Let  x : and  y  be two non-zero vectors in . The matrix is diagonalizable if≠0.(B) If A is symmetric and positive definite, then , is positive semi-definite.(C) For u,n ∈ such that =, where ⊕ is theoperation of direct sum of two vector spaces.(D) For the real matrix Q = , where u ∈ , we have det(Q) + rank(Q) = n.(E) None of the above is true.

11.Consider the first order differential equation xy'(x) = y(x) - with initialcondition y(2) = 0. Which of the following statements is/are true?(A) It is a linear differential equation for the dependent variable y.(B) y(x) is a parabolic function of x.(C) y'(x) =-x.(D) y"(x) =.(E) None of the above is true.

12. For the following homogeneous second order linear differential equation(x2-1)y"(x)-2xy''(x)+2y(x)=0for x ＞ 1, given one solution y1(x) = x, the other linearly independent solution y2(x) canthen be derived by setting y2(x) = v(x)y1(x). Which of the following statements is/aretrue?(A)(x3+x)v"(x)+2v'(x)=0.(B) u(x) with u'(x) = corresponds to one of the possible solutions.(C) u(x) with u'(x) = corresponds to one of the possible solutions.(D) y(x) = 1 + x + x2 is one of the possible solutions.(E) None of the above is true.

13. Continue from Question 十=. Solve the following non-homogeneous second order lineardifferential equation(x2-1)y"(x)-2xy'(x)+2y(x)=x2-1for x ＞ I using the variation of parameters, i.e,, set the particular solution aswhere y1(x) and y2(x) are homogeneous solutions from Question 十二. Which of thefollowing statements is/are true?(A)(B)(C) (D)(E) None of the above is true.

14. The following second order system can be transformed into an equivalent first order system for some real matrix A by introducing the four functions x1(t)= x(t),x2(t)=x'(t),x3(t) = y(t) and x4(t) = y'(t). Which of the following statements is/are true?(A) (B)(C) -36 is one of the eigenvalues of A.(D) 8 is one of the eigenvalues of A.(E) None of the above is true.

15. Continue from Question +e. Find the particular solution of the second order system inequation (2) with initial conditions z(0) = 2, x'(0) = 12, y(0) = 1 and y'(0) = 6. Whichof the following statements is/are true?(A) .(B) (t)=2cos(6t) +2sin(6t).(C).(D) y(t) = cos(6t) + sin(6t).(E) None of the above is true.

16. Consider the following second order differential equation3t2y"(t) + sin(t)y'(t) - cos(t)w(t) =0. be two linearly independent Frobeniusseries solution for y(t) in the above differential equation when t ＞ 0. Assuming r1≥ r2and a0 = 1, which of the following statements is/are true regarding the solution y1(t)?(A) r1 =1.(B) (C) (D) (E) None of the above is true.