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103 年 - 103 國立中山大學_碩士班招生考試_電機系(甲、丁、戊、己組):工程數學甲#110234 

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1. (15%) Evaluate the following integral62faf1fef1eb2.jpg


2. (15%) Deine the Fourier transform ofa signal f(t) as62faf2312db81.jpg


(2) (7%6) Compute the quantity A given below 62faf26242d51.jpgwhere the Fourier transform of f1(t) is given in the following figure (a).


(b) (8%) Compute the quantity B given below62faf2ad4c7f7.jpgwhere the Fourier transform of f2(t) is given in the following figure (b).


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 62faf65e95a69.jpg. 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).


(a) (6%) Please continue the argument to derive the result rank
(A) = rank(R).
(接下來前段是背景知識介紹,之後才是提問)Insthq given62fafb773ca88.jpg, 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 =62fafbdeddfce.jpgb is obtained. It seems that we may summarize the above argument as the following statement: 

Given62fafc9c456b6.jpg, where all columns of A are assumed linearly independent, then solution to the equation Ax = 6 can always be computed from x =62fafc231b349.jpg, where Q and R are matrices obtained from the QR factorization of A.

 However, the simple example62fafcc3302ee.jpg shows that the summary is incorrect because, according to the summary, the solution is x =62fafce3018f5.jpg=0 and obviously it does not satisfy the original equation.

5.【題組】(b) (5%) What is the error (or are the errors) in the argument right before the summary to make it incorrect?


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。
 Let V be a vector space with dim(V) = n and an ordered basis E := [x1, ..,xn].Let F := [y1,... , Yn] be the ordered orthonormal basis generated from basis E by applying the Gran- Schmidt Orthogonalization Process. For any62fafea987b59.jpg, 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 : 62fafee02960d.jpg, ie. L is a linear operator mapping V into itself, and suppose that 62faff2c42b0d.jpg
Let's denote the matrix representation of L with respect to basis E by A.

【題組】(a) (6%) Write an equation to indicate the relationship between the two coordinate vectors [L(v)]E and [v]F.

7.【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?


5. (25%) Consider the following system of differential equations: 62fb0093a7fda.jpgwhere62fb00aedfcd7.jpg are constant coefficients.


(a) (10%) Let 62fb00d243489.jpg. Suppose62fb00e7c7134.jpg 4 0 and u ≡ 0. Find the solution of x1 and x2 for the initial conditions x1(0) = 62fb01343762c.jpg.


(b) (4%)Suppose 62fb015a8611a.jpg Find the range of k: such that the solution of x1 and x2 converges to zero for any initial condition.


(c) (5%) Suppose62fb01b82a48e.jpgFind the range of k such that the solution of at and ta exhibits oscillatory behavior for any nonzero initial condition.


(d) (6%) For the values 62fb01de4657c.jpg, calculate the steady-state response of62fb021d437c1.jpg.

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.

13.【題組】(b) (10%) Find a potential function f(x, y, z) for F(c, y, z). 


(c) (2%) Find62fb038fda3ee.jpg, where C is the straight line going from the points (2,2,1) to the point (1,-1,2).


7. (5%) Evaluate the following integral




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10 x


103 年 - 103 國立中山大學_碩士班招生考試_電機系(甲、丁、戊、己組):工程數學甲#110234-阿摩線上測驗

103 年 - 103 國立中山大學_碩士班招生考試_電機系(甲、丁、戊、己組):工程數學甲#110234