| 申論題 | 1. (7%) Find the Laurent series representation of a function |
| 申論題 | 2. (8%) Evaluate the following integral: |
| 申論題 | 3. (15%) Compute the Fourier transform |
| 申論題 | 4.(15%)下面的問題共有三個子題,只要寫出每個子題的答案即可(不需寫出計算過程),例如:
(a) β=1,γ=2。 Let a1 = [1 0 1 α]T,a2=[I β 2 2]T, and a3 = [-2 3 γ -4]T be three vectors in R4,
where a, β, and γ are three real parameters, and denote A := [a1 a2 a3] 【題組】 (a) (4%) Suppose a is not a positive integer. Find real β and γsuch that{a1, a2, a3} is a linearly dependent set. |
| 申論題 | 【題組】 (b) (5%) Now let a = 2, β = -1,γ = -5, and let x be a nonzero vector in the null space N(A) of A. Find the value of k to satisty ||x|| 1 + 2||x||∞ + k||x||2 = 0. |
| 申論題 | 【題組】 (c) (6%) Now let a = 2, β = -1, γ= -5, and let d denote the distance between vector [1 4 0]T and R(AT), the range space of AT. Compute the value of d. |
| 申論題 | 5.(10%)下面的問題共有二個子題,只要寫出每個子題的答案即可(不需寫出計算過程),例如:
(a)θ=30或θ=π/6。
【題組】 (a) (4%) Compute the angle θ, taken value in [0, π /2), between 1 and x. |
| 申論題 | 【題組】(b) (6%) Find a vector u(x) in C[0, 1], so that {1, u(x)} forms an orthonormal basis for S. |
| 申論題 | 6. (20%) Consider the following system of differential equations:
【題組】 (a) (5%) Suppose u ≡ 0 and the equations are driven by non-zero initial conditions. Determine the
conditions on the coefficients |
| 申論題 | 【題組】 (b) (10%) Let the initial conditions be equal to zero. For the values |
| 申論題 | 【題組】 (c) (5%) For the values |
| 申論題 | 7. (15%) Consider the region R enclosed by the x-axis, x = I and y = x3, as illustrated below 【題組】 (a) (3% + 2%) Find the flux of |
| 申論題 | 【題組】 (b) (10%) Find the flux of |
| 申論題 | 8.(10%) Consider the following Lyapunov equation XA+ATX+Q=0 where A is a (n-dimensional) real square matrix, and X, Q are real symmetric matrices. 【題組】 (a) (5%) Suppose all eigenvalues of A have negative real parts. Show that X = |
| 申論題 | 【題組】(b) (5%) Suppose Q is positive definite and the Lyapunov equation has a positive definite solution X. Show that all eigenvalues of A have negative real parts. |
| 申論題 | 1. (15%) Evaluate the following integral |
| 申論題 | 2. (15%) Deine the Fourier transform ofa signal f(t) as 【題組】 (2) (7%6) Compute the quantity A given below
|
| 申論題 | 【題組】 (b) (8%) Compute the quantity B given below |
| 申論題 | 3.(11%)下面的問题共有二個子題,(1)子題要清楚地寫山證明,(b)子題只要簡短扼要地回答
提問即可。
Let A be any matrix in 【題組】 (a) (6%) Please continue the argument to derive the result rank(A) = rank(R).
Given However, the simple example |
| 申論題 | 【題組】(b) (5%) What is the error (or are the errors) in the argument right before the summary to make it incorrect? |
| 申論題 | 4.(14%)下面的問題共有二個子題,只要簡短扼要地回答提問即可,不須寫出答案背後的推
導。 【題組】(a) (6%) Write an equation to indicate the relationship between the two coordinate vectors [L(v)]E and [v]F. |
| 申論題 | 【題組】(b) (8%) Obviously, matrix T' relates to the two bases E and F. What conclusions about vectors in basis E and/or in basis F can be drawn if the matrix Tis known to be diagonal? |
| 申論題 | 5. (25%) Consider the following system of differential equations:
【題組】 (a) (10%) Let |
| 申論題 | 【題組】 (b) (4%)Suppose |
| 申論題 | 【題組】 (c) (5%) Suppose |
| 申論題 | 【題組】 (d) (6%) For the values |
| 申論題 | 6.(15%) Let F(c, y2Z) = (y +y2z)i +(2-z+2xyz)j + (-y +xy2)k 【題組】(a) (3%) Verify that F' is conservative. |
| 申論題 | 【題組】(b) (10%) Find a potential function f(x, y, z) for F(c, y, z). |
| 申論題 | 7. (5%) Evaluate the following integral |
| 申論題 | 6.(15%) Let F(c, y2Z) = (y +y2z)i +(2-z+2xyz)j + (-y +xy2)k 【題組】 (c) (2%) Find |
| 申論題 | Problem 1 (20%)
ป็น 8น
Let u be a solution to the heat equation: 【題組】 (a) (10%) Define the thermal energy is constant in time; i.e., |
| 申論題 | 【題組】 (b) (10%) Let f(x) = cos(πx). Find the solution u. |
| 申論題 | Problem 2 (25%)
Let F = (y2 + axz + yz)i +(x2+ bcy +xz)j +(x2 + cyz+ wy)k. 【題組】(a). (10%) Find the values of the constants a,b, c for which F is conservative. |
| 申論題 | 【題組】 (b). (15%) For the values found in (a), ind a surface S with the following property: the path integral |
| 申論題 | Problem 3 (13%)
【題組】 (a) Suppose ItaA is nonsingular, thus, for any nonzero scalar a, Ωa := |
| 申論題 | 【題組】 (b) If Ω1 , that is |
| 申論題 | 【題組】 (c) If Ωa. is an idempotent matrix, then what are all possible values of det A ? (5%) |
| 申論題 | Problem 4(12%)
本問題共有(a)、(b)兩個子题,每個子題都只要寫出提問的答案即可(不須寫出答案背後的推
導)。 【題組】 (a) Write out the set |
| 申論題 | 【題組】 (b) Consider the inner product space V = |
| 申論題 | Problem 5 (15%) Let C be a circle |z|=2 described in the counterclockwise direction. 【題組】 (a)(5%) Compute the following integral |
| 申論題 | 【題組】 (b)(10%) Suppose the answer you obtained in Part (a) is jnt . Use Part (a) to evaluate |
| 申論題 | Problem 6(15%)
Define the Fourier transform of a signal |
| 99 | 1.Given a continuous-time periodic signal |
| 99 | 2.Consider a discrete signal x(n) = cos(2nr / N), where integer N is the fundamental period. Let a be
the coefficients of the discrete-time Fourier series of x(n). Then which of the following statements is
correct?
(A) a1=-1/2,a2=1/2.
(B) a1=a2=j/2
(C) |
| 99 | 3.Consider an LTI system whose impulse response is G(jω)=1/(a+ jω), a ＞ 0. Suppose that there is
an input signal X(jω)=1/(a+ jω). Assume that the output signal is y(t)= |
| 99 | 4.Consider the following three systems, where [n] or x(t) is the system input, y[n] or y(t) denotes
the system output, and
I. |
| 99 | 5. Let z =1- |
| 99 | 6.Which one of the following functions, where z = x+ jy is a complex variable, is analytic?
(A) f(2)= |
| 99 | 7.Let z be a complex number. Which of the following statements is correct?
(A) Log(z1/z2) = Log(z1)+ Log(z2), where Log(z) is the principal value of the complex logarithm.
(B) cos(j) is not a real value.(C) |
| 99 | 8. Let f(z)=z/(z2+9), and C be a circle|z - j2|=4 in counterclockwise direction. The evaluation of |
| 99 | 9. Let f(z)= |
| 99 | 10. The Laurent series of |
| 申論題 | 以下第11題到第13題中之所有的提問,都不需要高出推導過程,只要高出答案即可,答案正確 就得分。 【題組】 (a) (5%) Suppose that m=n = 3, A = |
| 申論題 | 【題組】 (b) (5%) When the equation Ax = b is unsolvable, we may consider the so-called least squares problen to find a set of solutions, having the least squares error, fom solving a normal equation. Suppose that rank(A) =k ＜min(m, n) and let A = BC be a full rank decomposition of A . Use the known matrices B, C, and b to describe the unique projection vector p of b onto R(A) with the least || b-pll2. |
| 申論題 | 12. (10%) Let f1 =x+a and f2 = x-a , 【題組】 (a) (4%) Denote the angle between f1 and f2 by θ. Find all possible values of a2such that θ = π/4. |
| 申論題 | 【題組】(b) (6%) Now set a = 1. Find functions g1and g2 such that {g1, g2} is an orthonormal set that satisfies Span(g1) = Span(f1) and Span(g1,g2) =Span(f1, f2). |
| 申論題 | 13. (10%) Consider a linear trainsformation L:P2 →＞R2 defined by L(p(x): 【題組】 (A) (4%) Find all possible values of β such that |
| 申論題 | 【題組】 (b) (6%) Suppose that |
| 申論題 | 以下第14題到第15題中之所有的提問,需要寫出推導過程或詳细說明理,答案正確但沒有推導過程或說明不正確,將酌扣分數或不給分。 14. (20%) Consider the following set of differential equations  【題組】 (A) (15%) Let u(t) ≡ 0 and the initial conditions be x1(0)=x2(0)= |
| 申論題 | 【題組】 (B) (5%) Let initial conditions be x1(0) = x2(0) = c1(0) = |
| 申論題 | 15. (10%) Evaluate the following integral |
| 99 | 【題組】 1.微分方程式 |
| 99 | 2.微分方程式 |
| 99 | 3.微分方程式 |
| 99 | 【題組】4.拉普拉斯轉换(Laplace transform)為線性轉换。 (A)是(B)否 |
| 99 | 5.令函數y(t)的拉普拉斯轉换為Y(s)·則函數 |
| 99 | 6.函數 |
| 99 | 【題組】7.複函數f(z)=(z+1)/z在原點之外的所有複平面上皆為解析(analytic)。 (A)是(B)否 |
| 99 | 【題組】8.複函數f(z)=sin z之絕對值會隨著z的虛部增大而發散。 (A)是(B)否 |
| 99 | 9. 定義Del操作子為: |
| 99 | 【題組】10.承上題,Vφ在φ之定義域上的任何封閉路徑積分皆為0。 (A)是(B)否 |
| 99 | 下面11-15題為單選,考慮微分方程式 【題組】 11.假設u(t)≡0, |
| 99 | 12.假設u(t)≡, |
| 99 | 13.假設u(t)≡0, |
| 99 | 14.考慮將前述方程式就 |
| 99 | 15.考慮將前述方程式就 |
| 99 | 下面16-23題為複選題, 【題組】16. 令F(x,y,z)=(y+ay2z)i+(bx-z+2xyz)j+(cy+xy2)k。下敘述何者正確? (A)有超過一組的(a,b,c值能讓F成為一個保守的向量場。 (B)只有一組(a,b,c)值能讓F成為一個保守的向量場。 (C)(a,b,c)=(-1,0,1)讓成為一個保守的向量場。 (D)(a,b,c)=(0,1,-1)讓F成為一個保守的向量場。 (E)(a,b,c)=(1,1,-1)會讓成為一個保守的向量場。 |
| 99 | 17.令 |
| 99 | 18. Consider the linear equation Ax = b, where A = [a1, a2, a3, a4 ] ∈ |
| 99 | 19. Consider the linear equation Ax = b with A ∈ |
| 99 | 20.Consider the linear mapping L:V - W. Let |
| 99 | 21. Given vectors x, y, z in IR" and matrices A, B, C in (B) (C) (A+B)(A -B) = A2 -B2 (D) If AC = BC and C is not the zero matrix, then A = B. (E) If AB equals the zero matrix, then BA also equals the zero matrix. |
| 99 | 22.Let A ∈ |
| 99 | 23. Let A ∈ |
| 申論題 | 以下第24到第25題需要簡明寫出計算過程,答案正確但沒有計算過程,將酌扣分數或不給分。第24題到第25題中z=x+jy代表複數,其中xy是實數而j=√-1。 【題組】 24.
(a) Let I be the circle |
| 申論題 | 【題組】 (b) Evaluate the integral |
| 申論題 | 【題組】 25.(a) Define |
| 申論題 | 【題組】 (b) Evaluate the integral
|
.
where C denotes a counterclockwise simple closed contour |z| = 3.
of a signum function f(t) defined as 
Each calculation step is required for obtaining the credit. 
.
f(x)g(x)dc and the nomm 
.
Denote S := span (1, x} as a subspace of C[0, 1].
where
are constant coefficients.
such that 
,
= -1, and
= -2, and u(t) = sin(t), calculate the
steady-state response of y(t)=x1(t)-x2(t).
= (1 + y2)j out of R through the two sides C1 (the horizontal
segment) and C2 (the vertical segment).
= (1 + y2)j out of the third side C3.
dτ
a solution to the Lyapunov equation.
where the Fourier transform of f1(t) is given in the following figure (a).
where the Fourier transform of f2(t) is given in the following figure (b).
. Then, since rank(A) = dim(R(A)), the dimension of range of A, and
R(A) = R(AAT), we have the result rank(A) = rank(AAT). Therefore, when replacing A by its QR factorization, we get rank(A) = rank(QRRTQT).
, instead of using the elementary row operations (i.e. the Gauss eliminations) to
manipulate the equation, we may also apply the QR factorization to the equation to get QRx = b,
which implies further QT QRx = QTb. Since QTQ = In, it gives Rx = QTb. Thus, according to the
result of (a), when all columns of A are linearly independent, the square matrix R is nonsingular and
so the solution x =
b is obtained.
It seems that we may summarize the above argument as the following statement: 
, where all columns of A are assumed linearly independent, then
solution to the equation Ax = 6 can always be computed from x =
, where Q and R are
matrices obtained from the QR factorization of A.
shows that the summary is incorrect because, according
to the summary, the solution is x =
=0 and obviously it does not
satisfy the original equation.
, let [v]Eand [v]F denote the coordinate vectors
of v with respect to bases E and F, respectively. Let T denote the transition matrix from basis E to
basis F. Let L : 
, ie. L is a linear operator mapping V into itself, and suppose that
where
are constant coefficients.
. Suppose
4 0 and u ≡ 0. Find the
solution of x1 and x2 for the initial conditions x1(0) = 
.
Find the
range of k: such that the solution of x1 and x2 converges to zero for any initial condition.
Find the
range of k such that the solution of at and ta exhibits oscillatory behavior for any nonzero
initial condition.
,
calculate the steady-state response of
.
, where C is the straight line going from the points (2,2,1) to the point
(1,-1,2).
with boundary conditions: u(0, x) = f(x), O ≤ s ≤ I, and
=0, 0≤t＜∞. 
u(t,x)da. Show that under the above assumptions, T(t)
f(x)dr, for all t≥0.
F ㆍ dr is equal to O for any two points P, Q (connected by any curve C) on the surface S.
, and let 
(A) denote the set of all eigenvalues of A .
is a
well-defined
matrix. Then we know from knowledge of eigensystem of a square matrix that,
corresponding to any
. What is the mathematical relation betweenλ and
μ?(3%)
, is an orthogonal matrix, then what mathematical relation between A
and ATcan be derived? (5%)
,where N(●)and
R(●) indicate the null space and the range of a matrix, respectively. (5%)
, where ＜A,B) :=tr(ATB) for A and B in 
.
Describe Sh as the span of a set of orthonormal vectors in 
, n is a positive integer
dt, and its inverse Fourier
ransform is 
. It is already known that the Fourier transform of signal
x(i)=sin(ai) /(πi) is
and 9
.Compute the Fourier transform of the signal
(B) 
(C) 
(D)a0＜5,ω0＜2,a5＞0.
(E) None of the above statements are correct.
(D) 
(E) None of the above statements are correct.
, where u(t) is a
unit-step signal. Then which of the following statements is correct?
(A)β+y=-a
(B) βy=-a
(C) β-y=t+a
(D) βy=at
(E) None of the above statements are correct.
Which of the following statements is correct?
(A) I is time-invariant, II is linear, Ill is causal.
(B) I is memoryless, II is causal, III is nonlincar.
(C) I is stable, II is linear, III is memoryless.
(D) I is causal, II is memoryless, III is linear.
(E) None of the above statements are correct.
. Assume that 
, whereθ denotes the principal argument. Then which of the
following statements is correct?
(A) r =2,θ =π/5
(B) 
(C) 
(D) 
(E) None of the above statements are correct.
, where 
=x-jy.
(B) f(z) = xy+jx
(C)f(z)=x2ーjy2 (D) f(z)=x2+y2+j2xy
(E) None of the above statements are correct.
, n=0,±1,±2,......(D) z2+2z-ez= 
,where exp(2)=ez.
(E) None of the above statements are correct.
. Then which of the following statements is correct?
(A) a＜0,β＞0
(B) a ＞0,β＞0
(C)a=0,1＜β＜4
(D) a＞0, -3＜β＜4
(E) None of the above statements are correct.
, and C be the right-hand half of the circle|z|=2 form z = j2 to z =-j2. Compute the
value of S. f(z)dz =a+ jB. Then which of the following statements is correct?
(A) a=0,-15＜β＜0
(B) a＞0,0＜β＜15
(C)-12＜a+β＜l2
(D)-2＜aβ＜-12
(E) None of the above statements are correct
is 
Which of the following statements is correct?
(A) aβ=1/6
(B) aβ=1/12
(C) a+β=5/6
(D)a+β=1/3
(E) None of the above statements are correct. 
.
. Find the set of all solutions to the
-4 2
equation Ax = b if it is consistent. Otherwise, find vector p to solve 
and, moreover
peR(A)
compute the value of 
.
, be two vectors in the vector space C[0, 1] with inner
product ＜f,8＞:=
.
， for every p(x)
, with β＞1.
, the inverse of L, does not exist.
exists. Find the matrix representation of 
for P2 and R2, respectively. 
.Find
the solutions of the differential equations.
, and u(t) be the unit step
function. Does the solutions of the differential equations converge to constant values as time
approaches infinity? Justify your answers.
為非線性方程式。
(A)是(B)否
有三個平衡點。
(A)是(B)否
的解,不管初值為何,都會收斂到0。
(A)是(B)否
的拉普拉斯轉换為s2Y(s)。
(A)是(B)否
的傅立葉轉換(Fourier transform)為 
(A)是(B)否
。若為具有連續一二偏導函數的連續純量場函數,則x(Vφ)≠0除非φ是常數函數或線性函數。
(A)是(B)否
,並回答以下第11至15题。
。下列哪一組初值 
所對應的解不是z(t)≡0, 
。
(A)(0,0)(B)(π,0)(C)(0,π)(D)(0,一π)
。將前述方程式就 
線性化後之線性方程式,滿足以下哪個敘述?
(A)若b=0,則任何初值對應的解皆會收斂到0。
(B)若b=0,則有些初值對應的解皆會發散。
(C)若b=1,則任何初值對應的解皆會收斂到0。
(D)若b=-1,則有些初值對應的解皆會收斂到0。
。將前述方程式就 
線性化後之線性方程式,滿足以下哪個敘述?
(A)若b=0,則有些初值對應的解皆會收斂到0。
(B)若b=0,則任何初值對應的解皆會收斂到0。
(C)若b=1,則任何初值對應的解皆會收斂到。
(D)若b=1,則任何初值對應的解皆會發散。
線性化後之線性方程式。假設b=0,且該方程式之輸入項(forcing term)為單位步階函數。下列敘述何者為正確?
(A)該線性方程式的解會收斂到1。
(B)如該線性方程式的初值為(1,0),則方程式的解為sin t。
(C)該線性方程式的解會發斂
(D)該線性方程式的解會不斷。
線性化後之線性方程式。假設b=2,且該方程式之输入項(forcing trm)為sin t。下列敘述何者為正確?
(A)該線性方程式的解會收斂到一 
cost。
(B)該線性方程式的解會收斂到 sint。
(C)如該線性方程式的初值為(0,0),則方程式的解為sin t。
(D)如該線性方程式的初值為(0,1),則方程式的解為cos t。
。下列關於 
之敘述何者正確? (A)該矩陣是一個3x3的方陣。
(B)該矩陣在t≥0时,永為一個可逆矩陣。
(C)該矩陣(3,2)位置那一項為 
。
(D)該矩陣(1,2)置那一項為 
。
(E)當t→∞時,該矩陣收斂為0矩陣。
and a1, a2, a3, a4 are
column vectors of A. Suppose a1 + a2 + a3 + a4 = b. Which of the following statements are true?
(A) The linear equation has exactly one solution.
(B) The linear equation has infinitely many solutions.
(C) No conclusion can be drawn about the number of solutions to the linear equation.
(D) The vectors a1, a2,a3, a4 are linearly dependent.
(E) rank([A,b]) = rank 
. Which of the following statements are true?
(A) If rank 
= m, then there exists at least one solution.
(B) If rank 
= n, then there exists exactly one solution.
(C) Ifrank 
= n, then the column vectors of A are linearly independent.
(D) Ifn ＞ m, then there exists at least one solution.
(E) Ifm ＞ n, then there exists at most one solution.
be the zero vectors in V and W, respectively.
Which of the following statements are true?
(A) The condition L(V1) = L(V2) implies V1 = V2.
(B) For any w ∈ W, there exists v ∈ V such that L(V) = W.
(C) If L is one-to-one, then L(V) = 
impliesv = 
(D) IfV1,V2,..., 
are linearly independent, L(V1), L(V2),.., 
are also linearly independent.
(E) The condition G1V1:+C2V2+...+ 
= 
implies c1L(V1)+c2L(V2)+⋯..+ 
= 
. Which of the following statements are
true?
(A) 
, 
denote the column space of A, 
denote the null space of A,
, there exists u∈ 
and v ∈ 
such that x = u + v.
and v ∈ 
. Then 
= 0.
+ dim 
= n
, there exists x ∈ 
such that y = 
.
, then y is the zero vector in 
.
and x ∈ 
. Which of the following statements are true?
share the same eigenvalues and eigenvectors.
.
＞ 0.
oriented positively. Evaluate the integral
fatid
