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.

1. When using the Gaussian elimination to solve a linear equation Ax = b, elementary row operations (or multiplication by elementary matrices) are applied to the augmented matrix [A b]. Actually, there are many places in linear algebra where such a technique plays its role； e.g., the null space of A can be determined by setting b = 0. Which of the following cannot be determined by applying such a technique? (A) the range of A (B) the inverse of A (if it is nonsingular) (C) the QR factorization of A (D) the LU factorization of A (if it is square) (E) the determinant of A (if it is square).

2. The linear combination of a set of vectors is an essential element in linear algebra. We say a set V is invariant under linear combination if the implication "∀n ∈ and any set of vectors {v₁, ..., } ⊂ V ⇒ the set of all linear combinations {c₁v₁ + ... + } ⊂ V" holds. And we say a mapping L defined on a set X is invariant under linear combination if the form of linear combination is unchanged under L, or more precisely the statement ", and any set of vectors {x₁, ..., } ⊂ X, the identity L(c₁x₁ + ... + ) = c₁L(x₁) + ... + holds" is true. Which one of the following statements related to linear combination is false (A)Let S be a subset of a vector space V. Then S is a subspace of V if S is invariant under linear combination.  (B) Let {V₁, ..., } be a set of k subspaces of a vector space W and denote span{V₁, ..., } as the set of all linear combinations of the form c₁v₁ + ... + with each vᵢ chosen freely from . Then span{V₁, ..., } is also a subspace of W with dim(span{V₁, ..., }) = dim(V₁) + ... + dim()  (C) Let A and B be two matrices and denote C := AB. Then each column of C is a linear combination of all columns of A, and so rank≤ rank is implied. (D) A mapping L between two vector spaces is a linear transformation if and only if it is invariant under linear combination. (E) Let (V, (. ,. )ᵥ) be an inner product space. Then (. , .)ᵥ is invariant under linear combination at either one of its two arguments.

3. Consider the system represented by the differential equation ẍ+ẍ+kx. Which of the following is true? (A) the system is critically damped when k = 1 (B) the system is underdamped when k = 1/2 (C) the system is overdamped when k = 0.3 (D) the damping ratio of the system is increased when increasing k (E) none of the above

4. Consider the differential equation ẍ + bẋ + kx = cos(ωt) and its sinusoidal solution . Which of the following is true? (A) for a fixed b, the amplitude of is maximized when k = ω² (B) for a fixed b, the maximal amplitude of is 1/b (C) for fixed b and k, the amplitude of always grows as ω increases (D) the period of is 2πω (E) the period of depends on b and k.

5. Consider the differential equation tẋ + x = cos(ωt). Which of the following is true? (A) the general solution is sin(ωt) + C, C is a constant. (B) the particular (forced) solution is periodic with period ω (C) the solution is NOT bounded when t grows to infinity (D) If x(π/ω) = 1, then the solution is (sin(ωt) + π)/(ωt) (E) none of the above

複選題6. Consider a system whose dynamics is governed by a linear constant-coefficient ODE. Suppose the impulse response of this system is . Which of the following is true? (A) the system is order (B) the characteristic equation of the system has a pair of complex roots (C) the unit step response has oscillatory behavior (D) the unit step response does NOT converge to a constant value (E) the unit step response is equal to /3

7. Consider the heat equation , ∀0 ＜ x ＜ 1, t ＞ 0 u(0,t) = u(1,t) = 0 ∀t ＞ 0 u(x, 0) = f(x), ∀0 ＜ x ＜ 1 (A) the heat equation is nonlinear (B) without considering the boundary condition, the general solution is (C) suppose f(x) = 7 sin(3x), then the solution is u(x,t) = (D) all of the above are true (E) none of the above is true

8. Define the del operator ∇ := (A) ∇(∇ × F) = 0, where F is a vector field with continuous first and second derivatives. (B) ∇(F × G) = G ⋅ (∇ × F) – F ⋅ (∇ × G), where F and G are two smooth vector fields (C) ∇ ⋅ (φF) = ∇φ ⋅ F + (∇ ⋅ F), where φ is a smooth scalar field and F a smooth vector field. (D) all of the above are true (E) none of the above is true

9. The Fourier transform of the function f(t) = u(t-2) is F(jω) = e＜sup＞aω (b+jc), where u(t) is an unit-step function. Then which of the following statements is correct? (A) a=-2, b=1, c=0. (B) a = 2, b =1, c=0. (C) a＞0, b＞1, c=0. (D) a＜2, b≠1, c≠0. (E) None of the above statements are correct.

10. Given that f(t) has the Fourier transform F(jω). Let f1(t) = f(-3t-6), and let the Fourier transform of f1(t) be F₁(jω) = . Then which of the following statements is correct? (A) a=3, b=3, c=2. (B) 0＜a＜1, -1＜b＜0, 0＜c≤2. (C) a = -3, b=1/3, c=3. (D) a＞0, b＞0, c＜0. (E) None of the above statements are correct.

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