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104年 - 104 國立中山大學_碩士班招生考試_電機系(乙組):工程數學乙#110518
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題組內容
Problem 1 (25%) Consider the differential equation:
, with initial condition x(0) =x
0
.
(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?
其他申論題
(七)研究倫理的思考(3分);
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(八)確保研究品質的方法(5分)。
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(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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(a) What are all possible values of k so that S is a subspace of V := ? (2%)
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(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%)
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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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