[試題] 105-1 江衍偉 物理數學 期中考
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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※ 編輯: jerrysaikou (36.230.251.81), 01/18/2017 23:47:18