[試題] 107-1 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 :
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1. For the matrix A = | 0 3 0 | , find (a) its range R(A), (b) its null
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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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