課程名稱︰線性代數二 課程性質︰數學系大一必修 課程教師︰余正道 開課學院：理學院 開課系所︰數學系 考試日期︰2020年04月10日(五) 考試時限：11:20-11:45，共25分鐘 試題 : 1. Let V be a finite-dimensional inner product space. If T and U are linear operators on V, we wrtie T < U if U - T is a positive operator. Prove or disprove (a) (7%) If T < U and U < S, then T < S. (b) (7%) If T < 0 and U < 0, then TU > 0. 2. (8%) Show the following fourth order Hilbert matrix is positive definite. [ 1 1/2 1/3 1/4 ] H4 = [ 1/2 1/3 1/4 1/5 ]. [ 1/3 1/4 1/5 1/6 ] [ 1/4 1/5 1/6 1/7 ] 3. (8%) Let V be the polynomial function space of degree less than 2020 together with the 0-polynomial over |R equipped the inner product 1 (p|q) = ∫p(x)q(x)dx. 0 1 Define a linear functional L by L(p) = ∫p(x)dx for p∈V. For any 0 non-singular linear operator T on V, prove that there exists q∈V such that L(p) = (Tp|q). (Hint: You can use the result of Assignment 4 without proof but you need to write the statement which you use.) --

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