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

(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.

題組內容

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

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