九 、 Let v₁ and v₂ be the eigenvectors of matrix A corresponding respectively to eigenvalues λ₁ and λ₂, where  It is found that λ₁ > 1 > |λ₂l. With matrix A, ,

be a series of vectors related by . Given the initial condition x₀ = =

, which of the following statements is/are true?
(A)
(B)
(C)
(D)
(E) None of the above are true.

複選題五、Which of the following statements about matrix factorization is/are true? (A) If a matrix A is positive definite, then A has "an" LU factorization, A = LU, where the diagonal entries of U are positive.(B) Suppose that a matrix A = QR, where Q is an m x n matrix and R is an n X n matrix. If the columns of A are linearly independent, then R must be invertible.(C) Any factorization of a matrix A = , with matrices U, V square and positive diagonal entries in the matrix D, is called a singular value decomposition of A.(D) An n x n matrix A is positive definite if and only if A has "a" Cholesky factorization A = for some invertible upper triangular matrix R whose diagonal entries are all positive.(E) None of the above are true.

複選題六、 The Householder matrix H = is the orthogonalprojection matrix onto some nonzero vector , is a reflection matrix. Which of thefollowing statements is/are true?(A) Consider the 2-dimensional space, ie. n = 2. For vectors u, d, p,q   shown in thefigure below, we have H d = p.(B) H is a symmetric and orthogonal matrix.(C) Both linear systems A z = o and HAz = Ho are equivalent.(D) Let a = ≠0. Then llull2 = -2 llαll u1and Hα = (E) None of the above are true.

七、 Given029which of the following statements is/are true?(A) (B) (C) (D) det(F) = 20.(E) None of the above are true.

八、 Consider a 3 x 3 nonzero matrix A, and let W = . The leastsquares solutions xLs to Ax = b are illustrated in the following figure，where projwe b is theorthogonal projection of vector b onto vector space W. A minimun length least squares, solution EMurs is the one among the least squares solutions that has a minimum norm.Which of the following stateme ents is/are true?(A)(B) (C) (D) The minimum length least squares solution EMLLS to Ax = b is unique, and =projvxLs, where V = .(E) None of the above are true.

複選題十、Let , be a linear transformation on for a rotation by an angle θ about a unit vectoru. Specifically, we let the matrix for with respect to the standard basis S for bewhere is the coordinate vector of u with respect to S, c = cos(θ)and s = sin(θ). Furthermore, let A be the rotation matrix about the z-axis of Cartesiancoordinate system by an angle θ, i.e.,where be an ordered orthonormal basis for with and let be the changeof-basis matrix for changing basis from B to S. Which of the following statements is/aretrue?(A), n1n2 +b1b2 + u1u2 = 0 and n1n3 + b1b3 +u1u3 =0.(B) The coordinate vector of u with respect to basis B is .(C) (D) (E) The matrix for with respect to basis B is A.

複選題十一、 Solve for y(x) the first order differential equationxy'(x) - 4x2y(x) + 2y(x)ln(y(x)) = 0by the substitution u = In(y(x)). Which of the following statements is/are true?(A) It is a nonlinear ordinary differential equation for the dependent variable y.(B) It is a nonlinear ordinary differential equation for the new variable u.(C) There exists a solution y(x) satisfying the condition y(0) = 1.(D) There exists a solution y(x) satisfying the condition y(1) = 1.(E) None of the above are true.

複選題十二 、 The Cauchy-Euler equationx2y"(x)-4xy'(x)+6y(x)=0can be transformed into a constant coeficient equation y"(u) + by'(u) + cy(u) = 0 by thesubstitution u = In(x). Which of the following statements is/are true?(A) b= -5.(B) c=-6.(C) The solution y(x) exists only for x ＞ 0.(D) y(x) = C1x2 + C2x3 for some constants C1 and C2.(E) None of the above are true.

十三、Continued from Problem 十二 . Solve the non-homogeneous second order differential equa-tionx2y"(x)-4xy'(x)+6y(x)=x3by variation of parameters, i.e, set the particular solution as +u2(x)y2(x), where y1(x) and y2(x) are homogeneous solutions. With initial conditionsy(1) = 0 and y'(1) = 1 for the complete solution y(x), which of the following statementsis/are true?(A) The real valued solution y(x) exists only for t ＞ 0.(B) y(2) = 2.(C) 2y'(2) -3y(2) = 8.(D) y'(3) -y(3) = 4.(E) None of the above are true.

十四、 The first order systemcan be reduced into an equivalent second order system y"(t) = By(t) with y(t) =. Which of the following statements is/are true?(A) (B) (C) The eigenvalues of B are 9 and 25.(D) is an eigenvector for B.(E) None of the above are true.

十五、Continued from Problem +u. Find the partieular solution for the second order system y"(t) = By(t) with initial conditions . Which of the following statements is/are true? (A) (B) (C) (x1(t) +x3(t)) is an odd function in t.(D) (x1(t) -x3(t)) is an odd function in t.(E) None of the above are true .