110年 - 110 台灣聯合大學系統_碩士班招生考試_電機科:工程數學(C)#125211

1. Let

be the reduced row echelon form of a matrix A = [a1 a2 a3 a4 a5]. Which of the following statements is/are true? (A) a2 is the 4 × 1 zero vector. (B) The three column vectors a1, a4 and a5 are linearly independent. (C) The column space of R is the same as the column space of A. (D) Let B = [a1 a2 a3] be a submatrix of A. The reduced row echelon form of B is not always a submatrix of R. (E) None of the above is true.

2、Continue from Question. Which of the following statements is/are true?(A) The nonzero rows of R form a basis for col(A^T). (B) Let C be the reduced row echelon form of R^T. Then C^TC is the identity matrix. (C) Let E be a 3 × 5 matrix and be the reduced row echelon of E. We can obtain the intersection of the right null space of A and the right null space of E from R and D without knowing A and E. (D) Let Q be a 5 x 5 matrix and rank(Q) = 3. Then rank(AQ) = 3. (E) None of the above is true.

3、Let U and V be subspaces of Rⁿ and T be a linear transformation from Rⁿ to R^m. Which of the following sets is/are subspace(s) of Rⁿ?(A) {a + b ∈ Rⁿ : a ∈ U and b ∈ V}. (B) {a ∈ Rⁿ : a ∈ U or a ∈ V}. (C) {a ∈ Rⁿ : a ∈ U and a ∈ V}. (D) {a ∈ Rⁿ : T(a) = 0}. (E) None of the above is true.

4、Let A, B, and C be n×n matrices for some positive integer n. Which of the following statements 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 = I_n, then B = C. (C) If det( AB ) ≠ 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 ∈ R^m✖R, with the operations of matrix addition and multiplication of a matrix by a real scalar. Let T be a linear transformation from V to V, and {B₁, B₂,..., B_n} be a basis for V. Suppose U is a linear transformation from V to Rn given by U(c₁B₁ + c₂B₂ + ... + c_nB_n) =. Which of the following statements is/are true?(A) Let {A₁, A₂,..., A_k} be linearly independent over R. The set {U(A₁), U(A₂),…, U(A_k)} can be linearly dependent over R. (B) Suppose the dimension of the range space of T is k over R. Then {T(B₁), T(B₂),……, T(B_k)} are linearly independent over R. (C) n = m. (D) Let c = U(B₁) and a = U(T(B₁), then aTa = cTc. (E) None of the above is true.

6、For the matrix which of the following statements is/are true? (A){rank}( A ) = 3. (B) The sum of eigenvalues of Ais 6. (C) A is similar to B = (D) The system of linear equations Ax = [-1 1 1]^T \end{bmatrix} has a solution. (E) None of the above is true.

7、For a non-zero matrix A∈R^m✕n, define the maximum Frobenius norm||Ax|| over all unit-norm vectors x. Which of the following statements is/are true? (A)σ² is the maximal eigenvalue of AA^T. (B) If x₀ is an optimal solution to equation (1), i.e., ||Ax₀||= σ, then there exists a vector x1 such that Ax₁ = 0 and x₁^Tx_0≠0. (C) Assume m = n and A is nonsingular. Then. (D) Assume m = n. Then I_n + A is singular only if σ≥ 1. (E) None of the above is true.

8、Let V together with the inner product (•, •) v be an inner product space over R. For a
collection of N linearly independent vectors {w1, ..., w_N} in V, let M = [m_i，,j] ∈R^{N✖ N}be
the corresponding Gram matrix, i.e., the (i, j)-th entry of M is given by m{i_，,j} = (wi, wj)V.
Define the following set of real-valued functions f: V→R

which is a vector space over R when combined with standard addition and scalar multiplica-
tion of real-valued functions. For any
in the vector space F, define

Which of the following statements is/are true?(A) M is positive definite.(B) If m_1,1 = 2, m_2,2 = 1, and m_1,2 = -, then ||w₁ - w₂||V = 4, where ||\cdot||_V is the vector norm induced by the inner product(•, •) v.(C) (f, g)_F is an inner product for elements f, g∈ F.(D) Given g(y) = (v1, y)V, we have (f, g)_F = f(w1) for all f∈ F.
(E) None of the above is true.

9、Which of the following statements is/are true?(A) Let x and y be two non-zero vectors in Rᵀ. The matrix xyᵀ is diagonalizable if
yᵀx ≠ 0.(B) If A ∈ Rⁿˣⁿ is symmetric and positive definite, then A + A⁻¹ - 2Iₙ is positive semi- definite.(C) For u, v ∈ Rⁿ such that uᵀv ≠ 0, Rⁿ = (span{u})^⊥⊕span{u}, where ⊕ is the
operation of direct sum of two vector spaces.(D) For the real matrix Q = Iₙ - uuᵀ, where u ∈ Rⁿ, we have det(Q) + rank(Q) = n.
(E) None of the above is true.

10、Let P ∈ Rⁿˣⁿ be the orthogonal projection matrix, with respect to the Euclidean inner product, of vectors in Rⁿ onto col( A ) for some nonzero matrix A ∈ Rⁿˣᵏ. Which of the following statements is/are true?(A) tr(PᵀP) ≥ rank(P).(B) P can have an eigenvalue larger than 2.(C) P²x ≠x for all x ∈ Rⁿ.(D) If k ＜ n and b ∈ Rⁿ, the system of linear equations Ax = b has a unique least squares solution.
(E) None of the above is true.

11、Consider the first order differential equation xy'(x) = with initial condition 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) = 0
for x ＞ 1, given one solution y₁(x) = x, the other linearly independent solution y₂(x) can then be derived by setting y₂(x) = v(x)y₁(x). Which of the following statements is/are true?(A) (x³ + x)v''(x) + 2v'(x) = 0. (B) v(x) with v'(x) =corresponds to one of the possible solutions. (C) v(x) with v'(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 十二. Solve the following non-homogeneous second order linear differential equation
(x² - 1)y''(x) - 2xy'(x) + 2y(x) = x² - 1
for x ＞ 1 using the variation of parameters, i.e., set the particular solution as
y_p(x) = u₁(x)y₁(x) + u₂(x)y₂(x)
where y1(x) and y2(x) are homogeneous solutions from Question 十二. Which of the following statements is/are true?(A) (B) u'₁(x) = -(x² + 1) (C)u'₂(x) = (D) u'₂(x) = x (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 十四. Find the particular solution of the second order system in equation (2) with initial conditions x(0) = 2, x'(0) = 12, y(0) = 1 and y'(0) = 6. Which of the following statements is/are true?(A) x(t) = 2e^6t.(B) x(t) = 2 cos(6t) + 2 sin(6t).(C) y(t) = e^6t.(D) y(t) = cos(6t) + sin(6t).
(E) None of the above is true.

16、Consider the following second order differential equation
3t²y''(t) + sin(t)y'(t) - cos(t)y(t) = 0.
Let y1(t) = t^r1be two linearly independent Frobenius series solution for y(t) in the above differential equation when t ＞ 0. Assuming r₁≥r₂ and 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.

17、Continue from Question 十六. Assuming b₂ = 2, which of the following statements is/are true regarding the solution y₂(t)?(A) (B) b₁ = 0 (C) (D) (E) None of the above is true.

18、Consider the following differential equation for the function y(t) defined for t ≥ 0

where u(t) is the usual unit-step (Heaviside) function and where * is the usual convolutional operation of two functions. Assume y(t) = 0 for t ＜ 0. Given the initial condition y(0) = 1, which of the following statements is/are true regarding the Laplace transform Y(s) = {y(t)} of solution y(t) in the above equation?(A) Y(s =) = −6(B) Y(s = 1) = 1(C) Y(s = 2) = (D) Y(s = 3) =
(E) None of the above is true.

19、Continue from Question 十八. Which of the following statements is/are true regarding the solution y(t)?(A) y'(1) = 1 (B) y'(π) = 1 (C) y"(1) = -1 (D) y"(π) = 0 (E) None of the above is true.

20、Consider the following partial differential equation for the bivariate function v(x, t)

subject to conditions

Express the solution v(x, t) in the following form

for some constants an, bn, pn, qn, rn ∈ R satisfying 0 ≤ r₀ ＜ r₁ ＜ r₂ ＜ ... and ＞0
for n = 1, 2, ... Which of the following statements is/are true?(A) a₀ = - (B) b₁ = (C) p₃ = (D) r₄ = (E) None of the above is true.