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 following
statements 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 vector
x₁ 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.

3.Let U and V be subspaces of and T be a linear transformation from to . Whichof the following sets is/are subspace(s) of ?(A) (B)(C)(D) (E) None of the above is true.

複選題4.Let A, B, and C be n x n matrices for some positive integer n. Which of the followingstatements is/are true?(A) det( A ) = 0 implies det(R) == 0, where R is the reduced row echelon form of A.(B) If AB = CA - , then B = C.(C) If det(AB) 7 0, then A is invertible.(D) rank(AB)=rank(BA).(E) None of the above is true.

5. Let V be the vector space of all A ∈ , with the operations of matrix addition andmultiplication of a matrix by a real scalar. Let T be a linear transformation from V toY, and (B1,B2,... ,] be a basis for V. Suppose U is a linear transformation from D to given by U(c1B1 + c2B2 +.... +) = Which of the followingstatements is/are true?(A) Let(A1, A2,... ,] be linearly independent over R. The set ｛U(A1),U(A2),... ,｝can be linearly dependent over .(B) Suppose the dimension of the range space of T is k: over IR. Then ｛T(B1),T(B2),... ,｝are linearly independent over .(C) n=m.(D) Let  c = U(B1) and  a = U(T(B1), then (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.

10. Let P ∈ be the orthogonal projection matrix, with respect to the Euclidean innerproduct, of vectors in onto col( A ) for some nonzero matrix A ∈ . Which of thefollowing 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 squaressolution.(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.