題組內容

Problem 2 (20%) 本問题由兩個小題所組成,此兩小題共含有(a)~(f)六個子題；每個子題都只要寫出提問的答案 即可(不須寫出答案背後的推導)。
●(5%) Consider the set S:=63087db0867a7.jpg

(b) Consider the inner product space (S,〈●，●〉) with 〈A,B〉:=tr(ATB) for A and B in S. Find an orthogonal basis for S. (3%)

其他申論題

(a) (10%) Show that the solution of the equation is
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(b) (10%) Suppose a ＜ 0. Show that for any bounded u and any initial condition xo, the corresponding solution x is bounded.
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(c) (5%) Suppose a ＞ 0 and u(t) = sin(t). Find the initial condition xo such that the corresponding solution x is bounded. In this case, what is x?
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 (a) What are all possible values of k so that S is a subspace of V := ? (2%)
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(c) What is the condition on (A) so that I + aA is nonsingular? (2%)
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(d) Suppose I+aA is nonsingular, thus, for any nonzero scalar a, Ωa := is a well-defined matrix. Then we know from knowledge of eigensystem of a square matrix that, corresponding to any . What is the mathematical relation betweenλand μ? (3%)
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(e) If Ω1 , that is, is an orthogonal matrix, then what mathematical relation between A and AT can be derived? (5%)
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(f) If Ωa is an idempotent matrix, then what are all possible values of det A ? (5%)
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(a) Describe the set of all vectors [a β y]T that satisfy the two conditions , where N(●) and R(●) indicate the null space and the range of a matrix, respectively. (5%)
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(b) Now denote the set S := in terms ofsolution of (a). Write out the set S and discuss if the closure property of vector addition holds for set S, i.e. whether the implication "" holds for any P1 and P2. (1+5%)
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