110年 - [非官方正解]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.