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

(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(A^TB) for A and B in S. Find an orthogonal basis for S. (3%)

(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%)

(e) If Ω₁ , that is, is an orthogonal matrix, then what mathematical relation between A and A^T can be derived? (5%)

(f) If Ωa is an idempotent matrix, then what are all possible values of det A ? (5%)

(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%)

(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%)

(c) Consider the inner product space V = (,〈●_，●〉), where 〈A,B〉:= tr(A^TB) for A and B in . Describe S^⊥ as the span of a set of orthonormal vectors in . (7%)

(2) Now lot T = be a subspace of V and let P be any vector of S. What is the distance of P to T? (7%)

(a)(5%) Compute the following integral

n is a positive integer

(b)(10%) Suppose the answer you obtained in Part (a) is jnπ . Use Part (a) to evaluate

Problem 5 (15%) Define the Fourier transform of a signal f(V) as , and its inverse Fourier transform is. It is already known that the Fourier transform of signal x(t)=sin(at)/(πt) is and ). Compute the Fourier transform of the signal

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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