7. (5%) Evaluate the following integral

 

1. (15%) Evaluate the following integral

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

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

(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 given, 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: Given, 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. However, the simple example 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.

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

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

(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?

(a) (10%) Let . Suppose 4 0 and u ≡ 0. Find the solution of x1 and x2 for the initial conditions x1(0) = .

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

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

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