10. Let U, V, W be vector spaces and T: U →rightarrow V, S: V →rightarrow W be linear transformations. Also denote the set of all linear transformations from V to W by 
(V,W) (e.g. S 
(V,W)). Which one of the following statements is false? (A) The set (
(V,W), +, ‧), where (α ‧S1 + β ‧ S2)(v) =α ‧S1(v) +β S2(v) for any v
V, 
(V,W), and α ‧ β
R, is also a vector space with dim\mathcal
(V,W) = dimV + dimW.
(B) The mapping ST(.) defined as ST(u) := S(T(u)) for any u \in U is also a linear transformation from U to W.
(C) If both S and T are one-to-one, then their combination ST(.) is also one-to-one.
(D) If S: V →rightarrow W is invertible, then the mapping 
from W to V is also a linear transformation. (E) Suppose that the combination ST(.) is onto. Then at least one of S and T is onto.
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+ kx = 0. 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
+ kx = cos(ω t) 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 2ω (E) the period of 
depends on b and k.
+ x = cos(ωa t). Which of the following is true? (A) the general solution is sin(ω t) + 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(ω t) +π) / (ω t) (E) none of the above
. Which of the following is true? (A) the system is 
order (B) the system is 
order (C) the characteristic equation of the system has a pair of complex roots (D) one root of the characteristic equation is less than -5 (E) response to a step function does NOT converge to a constant value
(E)
= 4t - 3x ? (A) x(t) = constant (B) x(t) = -ct3 (C) x(t) = 
+ t (D) x(t) = 
+ 4t (E) none of the above
(A) the heat equation is nonlinear (B) without considering the boundary condition, the general solution is 
(C) suppose f(x) = 7sin(3x), then the solution is u(x,t) = 
(D) all of the above are true (E) none of the above is true
and any set of vectors 
the set of all linear combinations 
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 
X, the identity 
= 
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.
be a set of k subspaces of a vector space W and denote 
as the set of all linear combinations of the form 
, with each 
chosen freely from 
. Then spa
is also a subspace of W with 
=
≤ rank
is implied. 
is linearly independent if there exist coefficients 
, all of them are zero, such that 
= 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 
be the i th eigenvalue-eigenvector-pair of a square matrix. Then all 
's are distinct if and only if the set 
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(⋅) ∈ V. Which of the following statements are true? (A) Both 
are subspaces of V. (B) 
(C)
. (D) dimV = 
(E) Denote dist(f, 
) as the distance induced from the defined inner product between any f(⋅) ∈ 
and 
. Then dist((x+1)², 
) = √8/3.
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 
) 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.
be a basis of 
and F := 
⊂ 
with the property 
= 1,…,n uᵢᵀvⱼ = 1 when i = j, and 
= 0 when i ≠ j. Which of the following statements are true? (A) F is also a basis of 
, and the coordinate vector of any x ∈ 
with respect to base F is 
(B) Any x ∈ 
can be represented as x = c₁u₁ + ... + 
= 
for each i. (C) Denote the transition matrix from base E to base F by S. Then S(i, j) = 
, for i, j = 1,…, n. (D) The transition matrix from base F to base E can be described by (
), where 
denotes the coordinate vector of 
with respect to base E. (E) Denote U := 
and V := 
. Then Ux = λx for some x ≠ 0 iff 
, where the upper bar means to take the complex conjugate.
, let's consider the matrix 
= I − 
parameterized by a real scalar α. Which of the following statements are true? (A) 
= 
, i.e. 
is an idempotent matrix, if and only if α ∈ {0, 1, 2}. (B) 
= x for any x ∈ 
if and only if α ∈ {0, 1, 2}. (C) 
is singular if and only if α = 1. (D) For any x ∈
, 
, the orthogonal complement of w, if and only if α = 1. (E) For any x ∈ 
, 
= ‖H₁x‖₂, i.e. 
x has the smallest 2- norm if and only if α = 1.
be a nonzero matrix of size n×n with its real part and imaginary part being denoted by B and C, respectively. Moreover, let Ω be a symmetric matrix with A and C as its four blocks. Which of the following statements are true? (A) Matrix C has at least one real eigenvalues. (B) Let λ and μ be any two eigenvalues of B and C, respectively. Then λ + μ ≠ 0. (C) Any eigenvalue of A is also an eigenvalue of Ω. (D) Any eigenvalue of Ω is also an eigenvalue of A. (E) Let 
be an eigenvalue-eigenvector-pair of Ω. Then (λ, x+iy) is an eigenvalue- eigenvector-pair of A.
and for all functions f1, f2 (B) if L(f) = F(s), then
= sF(s) (C) if L(f) = F(s), then L(
⋅ f) = F(s)/(s+a) (D) if L(f) = F(s), then L(t² ⋅ f) = 
(E) if L(f) = F(s) and L(g) = G(s), then L(f * g) = (F * G)(s), where * denotes convolution operation
+ kx(t) = u(t). Suppose 
sin(2t) is a solution when u(t) = 4 cos(2t). Which of the followings are true? (A) m = 1 (B) b = 0 (C) k = 8 (D) 2tsin(t) is a solution when u(t) = 8cos(2t) (E) 
)sin(2t - 1) is a solution when u(t) = 4cos(2t - 1)
, where A = 
(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 equilibria 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.