109年 - 110 台灣聯合大學_碩士班招生_電機類：工程數學C#103784

一、 Which of the following sets is basis (are bases) for ? (A) .(B) (C) (D) (E)None of the above are true.

二、 Which of the following statements about the multiplicative inverse of a matrix is/are true?(A) A matrix A is called invertible if there exists a matrix B such that AB is an identity
matrix.(B) If a matrix is both diagonalizable and invertible, then so is its multiplicative inverse.(C) Suppose that matrices A of size n x n and D of size m x m are invertible and that
matrix C is of size m X fi； then the following identity is true
(D) Methods for finding the multiplicative inverse of a matrix include LU factorization,
Gaussian elimination, eigen-decomposition, and Gram-Schmidt process.(E) None of the above are true.

三、Which of the following properties of eigenvalue is/are true? (A) A scalar λ is an eigenvalue of matrix A if and only if λ is an eigenvalue of .(B) A matrix is positive semi-definite if and only if all of its eigenvalues are non-negative.(C) Every eigenvalue of a matrix A is also an eigenvalue of A².(D) If matrices A and B are similar, then they have the same eigenvalues.(E) None of the above are true.

四 、 Which of the following matrices is/are diagonalizable? (A) (B) (C) (D) (E) None of the above are true. 

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.

六、 The Householder matrix H = is the orthogonal
projection matrix onto some nonzero vector , is a reflection matrix. Which of the
following statements is/are true?(A) Consider the 2-dimensional space, ie. n = 2. For vectors u, d, p,q   shown in the
figure 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 u₁

and Hα = (E) None of the above are true.

七、 Given
029
which 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 least
squares solutions xLs to Ax = b are illustrated in the following figure，where projwe b is the
orthogonal 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.

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.

十、Let , be a linear transformation on for a rotation by an angle θ about a unit vector
u. Specifically, we let the matrix for with respect to the standard basis S for be

where 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 Cartesian
coordinate system by an angle θ, i.e.,

where be an ordered orthonormal basis for with
and let be the change
of-basis matrix for changing basis from B to S. Which of the following statements is/are
true?(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.

十三、Continued from Problem 十二 . Solve the non-homogeneous second order differential equa-
tion
x²y"(x)-4xy'(x)+6y(x)=x³
by 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 conditions
y(1) = 0 and y'(1) = 1 for the complete solution y(x), which of the following statements
is/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 system

can 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 .

十六 、For s ＞ 0, let F(s) be the unilateral Laplace transform of function f(t) given by

Which of the following statements regarding the values of f(t) is/are true?(A) (B) f(2) = 1.(C) f(4)=2.(D) f(8) = 3.
(E) None of the above are true.

十七 、 Consider the following periodic function

which has a Fourier series representation

for some . Which of the following statements is/are true?(A) (B) (C) (D) (E) None of the above are true.
m＞o

十八、 For the following second order differential equation
tx"(t)+(4t-2)x'(t)+(2t-4)x(t)=0
Let x₁(t)= be the two linearly independent
Frobenius series solutions for x(t) when x ＞ 0, where ri and r₂ are the zeros of the
corresponding indicial equation. Assume r₁ ≥ r₂ and ao = b₀ = 1. Which of the following
statements is/are true?(A) r₁ - r₂ is not an integer.(B) .(C) (D) .(E) None of the above are true.

十九、Continued from Problem 十i. The second order differential equation can be alternatively
solved by using Laplace transform. Assuming x(0) = 0 and fo = 1, which of
the following staterents is/are true about the values of x(t) and its unilateral Laplace
transform X(s) = ?(A) x'(0) = 1.

(B)x(1)=1.

(C) X(1) ＜1.

(D) Values of X(s) exists for all s ＞ -1.

(E) None of the above are true.

二十、Consider the following boundary value problem for the bivariate function u(x,t) defined
for 0 ≤x≤π and t≥0

Given the end-point and initial conditions
=0 and u(x,0)=x(π-x)
which of the following statements is/are true for the solution u(x,t) when it is expressed
 as
for some constants (A) (B) a₁=-1.(C) p₂ =-31.(D) b₁ =-1(E) None of the above are true.