課程名稱︰物理數學 課程性質︰選修 課程教師︰江衍偉 開課學院：電資學院 開課系所︰光電工程學研究所 考試日期（年月日）︰107/11/14 考試時限（分鐘）：170分鐘 是否需發放獎勵金：是 試題 : | 0 0 0 | 1. For the matrix A = | 0 3 0 | , find (a) its range R(A), (b) its null | 0 0 5 | space N(A), and (c) the pseudo-inverse A' of A. (Please write down the details of computation for part (c).) (12%) 2. (a) What is the Schwarz inequality? Write down its expression and descr- ibe its validity condition. (b) By using the Schwarz inequality, show that L^2[a,b] is a subset of L^1[a,b]. (c) Again by using the Schwarz inequality, show that if a function sequ- ence {fn(x)} converges with respect to the L^2 norm, then it converg- es to the same limit function with respect to the L^1 norm. (d) Show that an L^2 function can induce a distribution through the usual inner product. (20%) 3. Suppose that {Φ_i(x)}|_{i=1}^{\infty} is an orthonormal set and f(x) is a given function, all defined on [a,b]. (a) Find the expression of the generalized Fourier series of f expanded by {Φ_i}, and explain the meaning of this expansion. (b) Does the above Fourier series converge to f? In what sense? Discuss it briefly. (c) If Φ_i(x) are linearly independent but not orthogonal to each other, can we express f in terms of a series of Φ_i? Discuss it briefly. (18%) 4. (a) Solve the integral equation 2x^3 + 5x - u(x) + \int_0^1 (3x\zeta^2-4) u(\zeta)d\zeta = 0. (b) Verify your solution by substituting it back in the integral equation. (18%) 5. Can the inverse of a compact operator K, if it exists, be bounded? Explain briefly. (9%) 6. Discuss briefly the role that the Riesz representation theorem plays in establishing the theory of distributions. (9%) 7. Evaluate in the sense of distribution (a) lim(n\to\infty) cos[n^4(x+5n^9)] (b) lim(n\to\infty) n^11 cos[n^4(x+5n^9)] --

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