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題組內容

Problem 1 (25%) Consider the differential equation:, with initial condition x(0) =x₀.

(a) (10%) Show that the solution of the equation is

其他申論題

(五)研究設計(10分)；

(六)研究參與者(3分)；

(七)研究倫理的思考(3分)；

(八)確保研究品質的方法(5分)。

(b) (10%) Suppose a ＜ 0. Show that for any bounded u and any initial condition xo, the corresponding solution x is bounded.

(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?

(a) What are all possible values of k so that S is a subspace of V := ? (2%)

(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%)

(c) What is the condition on (A) so that I + aA is nonsingular? (2%)

(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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