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：電機資訊學院 開課系所︰電信工程學研究所 考試日期（年月日）︰105.11.9 考試時限（分鐘）：170 是否發放獎勵金：是 試題 : 4 0 0 1. For the matrix A= 0 2 0 , find (a) its range R(A), (b) its null space N(A), 0 0 0 and (c) the pseudo-inverse A' of A. (Please write down the details of computation for part (c).) [15%] 2. (a) Why do we need to define a metric d in a function space? Explain briefly. [3%] (b) The word "complete" has two different meanings in our textbook. Please give appropriate definitions and explain briefly. [6%] 3. (a) What is the Schwarz inequality? Write down its expression and describe its validity condition. [5%] (b) By using the Schwarz inequality, show that L^2[a,b] is a subset of L^1[a,b]. [5%] (c) Again by using the Schwarz inequality, show that if a function sequence {f_n(x)} converges with respect to the L^2 norm, then it converges to the same limit function with respect to the L^1 norm. [5%] (d) Show that an L^2 function can induce a distribution through the usual inner product space. [5%] 4. Assume that K:H→H is a compact linear operator defined on a Hilbert space H and {φ_n} is an infinite set of orthonormal functions in H. (a) Show that lim_n→∞ Kφ_n = 0. [8%] (b) Can the inverse of a compact linear operator, if it exists, be bounded? Explain briefly. [4%] 5. Solve the integral equation 7x^4 + 5 -u(x) + integral[0→1](x^2ξ^2-3)u(ξ)dξ = 0 , and verify your answer by substituting it back in the integral equation. [18%] 6. (a) Discuss briefly the role that the Riesz representation theorem plays in establishing the theory of distributions. [6%] (b) Note that the Dirac delta function is a continuous unbounded linear functional. Why can it be continuous but unbounded? Explain briefly. [6%] 7. Evaluate the following limit in the sense of distribution: (a) lim_n→∞ sin[n^7(x-8n^5)] (b) lim_n→∞ (n^16)sin[n^7(x-8n^5)] [14%] --

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