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.
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. 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.
. 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
, ∀0 ＜ x ＜ 1, t ＞ 0
(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
(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
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.
. 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.
dt is equal to α. Which of the following statements is correct? 
(A) α = 0 (B) α = 6√π (C) α = 3π (D) α = 3 (E) None of the above statements are correct. 
(B) f(z) = z Im z, Im z stands for the imaginary part of z (C) f(z) = x² + jy² (D)
(E) None of the above statements are correct.
f(z)dz is c + jd. Then which of the following statements is correct? (A) c ＜ 0, d ≤ 0 (B) c ＞ 0, d ＞ 0 (C) c = 0, 15 ＜ d ＜ 20 (D) c = 0, 1 ＜ d ＜ 10 (E) None of the above statements are correct.
(E) None of the above statements are correct.
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.
} 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.
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?
are subspaces of V.
) as the distance induced from the defined inner product between any f(﹒) ∈ V and 
. Then dist((x + 1)2, 
) = √8/3.
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.
(‧) 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
=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
(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.