14. Let z be a complex number. Which of the following statements is correct?
(A) │sin z │²is an unbounded function.
(B) │cos z│² is a bounded function.
(C) log(i²) = 2logi
(D)
(E) None of the above statements are correct.

15. Let f(z) = z⁵ + 2z³ + 3z² + 2, and let the number of zeros of f(z), counting multiplicities, inside the circle | z |= 2 be α. Which of the following statements is correct? (A) α = 3 (B) α = 5 (C) α is an even number (D) α = 1 (E) None of the above statements are correct.

16. Let f(z) = x2 + jy, and C is the path from z = 0 to z = 1 + j2 along the parabola y = x2. Compute the value of f(z)dz = a + jb. Then which of the following statements is correct? (A) a ＜ 0, b ≤ 0 (B) a + b = -2/3 (C) a + b = 2/3 (D) a ＞ 0, b ＜ 0 (E) None of the above statements are correct.

複選題17. The linear independence is without doubt one of the most important concepts in linear algebra. Which of the following statements about this concept are true? (A) The number of linearly independent columns of any matrix is equal to the number of linearly independent rows of the matrix. (B) A set of vectors {v1, ..., } is linearly independent if there exist coefficients c1, ..., , all of them are zero, such that c1v1 + ... + = 0. (C) The linear independence of a set of vectors is a necessary but not sufficient condition for them to be a basis of a vector space. (D) A square matrix is diagonalizable if and only if all its eigenvectors are linearly independent. (E) Let (λ1, x1) be the i th eigenvalue-eigenvector-pair of a square matrix. Then all λ's are distinct if and only if the set {x1, ..., } of eigenvectors is linearly independent.

複選題18. Let V = C[-1, 1] be the vector space of all continuous functions defined on [-1, 1], and let and denote, respectively, the set of all even and all odd functions in V. Moreover, define an inner product (f, g) := f(x)g(x)dx for any f(﹒), g(﹒) \in V. Which of the following statements are true? (A) Both  are subspaces of V. (B) (C) (D) dim V = (E) Denote dist(f, ) as the distance induced from the defined inner product between any f(﹒) ∈ V and . Then dist((x + 1)2, ) = √8/3.

複選題19. Which of the following statements are true?  (A) Given A and b of proper dimensions, when b ∉ R( A ) the linear equation Ax = b has no solution. However, the associated LSP (least squares problem) is always solvable and the solution is unique. (B) Let V be a vector space such that V = X ⊕ Y. Then only when X ⊥ Y, i.e. the two subspaces are orthogonal, can two projection mappings, say P: V → X and Q: V → Y, be defined with the complementary property P + Q = I, where I indicates the identity mapping on V.  (C) (continue from ( B )) The projections P and Q become orthogonal projections when X ⊥ Y. Moreover, once the bases for X and Y are chosen, their union forms a basis for V, and the two projection matrices associated with projections P and Q with respect to this set of bases are all symmetric.  (D) Any projection mapping is a linear transformation that is definitely onto, but may not be one- to-one.  (E) Consider the vector space C[-1,1] with an inner product (f, g) := f(x)g(x)dx. Then the set {u₁, u₂} with u₁ = 1/√2 and u₂ = (√6/2)x forms an orthonormal set in C[-1,1]. Moreover, the best least squares approximation to h(x) = x² by a linear function is = (√2/3)+(√6/4)x.

複選題20. Let (‧) denote the Laplace transform. Which of the followings are true? (A) (a₁f₁+a₂f₂) = a₁ (f₁) + a₂ (f₂), for all a₁, a₂ ∈  and for all functions f₁, f₂ (B) if (f) = F(s), then  = sF(s) (C) if (f) = F(s), then  = F(s)/(s+a) (D) if (f) = F(s), then (t² f) = (E) if (f) = F(s) and (g) = G(s), then (f ‧g) = (F*G)(s), where * denotes convolution operation

複選題21. Consider the homogeneous system =where ； (A) the system is linear (B) the system is time invariant (C) the system has more than one equilibrium (D) when a ＜ -4, any initial condition results in a solution which diverges to infinity (E) when -4 ＜ a ＜ -1, any initial condition results in a solution which converges to the origin

複選題22. Consider the autonomous system (A) the system is linear (B) the system has two equilibria (C) (x₁, x₂) = (2,-2), (-2, -2) are equiliria of the system (D) (x₁, x₂) = (-2, -2) is the only stable equilibrium of the system (E) (x₁, x₂) = (2, -2) is a saddle equilibrium of the system.

1.是從平均數為 µ,標準差為 σ² 的常態分配裡,抽取 m 個樣本,計算這 m 個樣本的平均數,然後放回母群,如此重複很 多次(譬如說做了 N 次),(A)O(B)X

2.則這些被計算出的 N 個平均數所構成的分配叫做「樣本平均數的抽樣分配」,(A)O(B)X

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