1. (15%) Evaluate the following integral

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

(a) (7%) 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 below where 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)In solving the linear equation Ax=b for a igiven , instead of using the elemnentary 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 QTQRx = 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 = o can always be computed from x = , where Q and R are matrices obtained from the @R factorization of A. However, the simple example shows that the summary is incorrect because, according to the summary, the solution is x=  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?

(c) (8%) What does the obtained solution x = mean? Give your answer necessary explanations.

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

(b) (7%) 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 T is known to be diagonal?

(c) (6%) Suppose now that and denote X := [x1,... ,xn].Let X = QR be the QR factorization of matrix X. Is there any relationship between matrices T, Q and R? If yes, write an equation to describe such a relationship. If no, give a brief explanation for it.

(d) (7%) Under what conditions on ayi for i, j = 1,... ,n will matrix A be an upper-triangular one?