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︰光電所 考試日期（年月日）︰2011/11/09 考試時限（分鐘）：2hr 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : 1. 1 -1 For the natrix A = [ ] , find (a) its adjoint A* , (b)its null space -6 6 N(A),(c) the null space of its adjoint,N(A*),(d)the pseudo-inverse A' of A , and (e) AA' , (f) compute A*AA' and explain the meaning of the result briefly. (18%) 2. (a) What is the Schwarz inequality ? Write down its expression and describe 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 sequence {f_n (x) } converges with respect to the L^2 norm , then if converges to the same limit function with respect to the L^1 norm.(18%) 3. (a) Why do we need to define a metric d in a function space? Explain briefly. (b) The word "complete" has two different meaning in our textbook. Please give appropriate definitions and explain briefly. (10%) 4. Consider a linear function T defined by Tf=f(0) , with f∈C [-1,1]. (a) Is T a bounded functional for f measured with the L^1 norm ? Explain briefly. (b) Is T a bounded functional for f measured with L^2 norm ? Explain briefly, (c) Is T a bounded functional for f measured with uniform norm? Explain briefly. (15%) 5. 1 (a) Solve the integral equation 5x^4＋2－u(x)+∫ (xξ^2－4)u(ξ)dξ=0 0 (b) Verify your solution by substituting it back in the integral equation.(15%) 6. (a) If N is a null space of a bounded linear functional defined on Hilbert space H and is a proper subset of H. Show that N',the orthogonal complement of N, is a one-dimensional space. (b) Discuss briefly the role that the Riesz representation theorem plays in establishing the theory of distributions (12%) 7. Evaluate (a) lim cos[n^6 (x+3n^3)] and (b) lim n^15 cos[n^6(x+3n^3)], in the n->∞ n->∞ sense of distribution. (12%) --

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