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題組內容

Problem 4 (15%) Let C be a circle |z| = 2 described in the counterclockwise direction.

(a)(5%) Compute the following integral
n is a positive integer

其他申論題

(a) Describe the set of all vectors [a β y]T that satisfy the two conditions , where N(●) and R(●) indicate the null space and the range of a matrix, respectively. (5%)

(b) Now denote the set S := in terms ofsolution of (a). Write out the set S and discuss if the closure property of vector addition holds for set S, i.e. whether the implication "" holds for any P1 and P2. (1+5%)

(c) Consider the inner product space V = (,〈●，●〉), where 〈A,B〉:= tr(ATB) for A and B in . Describe S⊥ as the span of a set of orthonormal vectors in . (7%)

(2) Now lot T = be a subspace of V and let P be any vector of S. What is the distance of P to T? (7%)

(b)(10%) Suppose the answer you obtained in Part (a) is jnπ . Use Part (a) to evaluate

Problem 5 (15%) Define the Fourier transform of a signal f(V) as , and its inverse Fourier transform is. It is already known that the Fourier transform of signal x(t)=sin(at)/(πt) is and ). Compute the Fourier transform of the signal

V. Essay Writing: Please write a well-organized essay in 250-300 words. As part of the effort to develop Taiwan into a bilingual nation by 2030, the EMI (English as a Medium of Instruction) program introduced by the Ministry of Education encourages universities to provide courses in English. Do you agree or disagree with this policy? Use specific reasons and examples to support your opinion.

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