110年 - 110 台灣聯合大學_碩士班招生_電機類：工程數學C#124898

1. Let
be the reduced row echelon form of a matrix A = . Which of the
following statements is/are true?(A) is the 4 x 1 zero vector.(B) The three column vectors , are linearly independent.(C) The column space of R is the same as the column space of A.(D) Let B = 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.
R=

2. Continue from Question -. Which of the following statements is/are true?
(A) The nonzero rows of R form a basis for col.(B) Let C be the reduced row echelon form of . Then is the identity matrix.(C) Let E be a 3 x 5 matrix and
[10110]
D = 01110
100001
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 and T be a linear transformation from to . Which
of the following sets is/are subspace(s) of ?(A) (B)(C)(D)
(E) None of the above is true.

4.Let A, B, and C be n x 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 - , 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 and
multiplication of a matrix by a real scalar. Let T be a linear transformation from V to
Y, 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 following
statements 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 matrix

which 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.

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.

8. Let V together with the inner product (‧,‧)y be an inner product space over . For a
collection of N linearly independent vectors [w₁ ,...,] in V, let M = be
the 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, 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 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 the
operation 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 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.

11.Consider the first order differential equation xy'(x) = y(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) 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 linear
differential equation
(x²-1)y"(x)-2xy'(x)+2y(x)=x²-1
for x ＞ I using the variation of parameters, i.e,, set the particular solution as

where y₁(x) and y₂(x) are homogeneous solutions from Question 十二. Which of the
following 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 in
equation (2) with initial conditions z(0) = 2, x'(0) = 12, y(0) = 1 and y'(0) = 6. Which
of 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 equation
3t²y"(t) + sin(t)y'(t) - cos(t)w(t) =0.

be 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 b₀ = 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 staterents is/are true regarding the Laplace transform
Y(s) = ((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 t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)
, 0 ≤ x ≤ 10, t ≥ 0
subject to conditions

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

for some constants ,  ∈ satisfying 0 ≤ r₀＜r₁＞r₂＜... ＞ 0 for n = 1, 2,... Which of the following statements is/are true?(A)(B)(C) (D)
(E) None of the above is true.