複選題

12. 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(⋅) ∈ 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.

8. Consider the heat equation (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

9. 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 " 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. (B) Let  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 = (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.

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 ‧ βR, 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.

複選題11. 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 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.

複選題13. Which of the following statements are true? (A) Given A and b of proper dimensions, when b ∉ R 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.

複選題14. Let E := { 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.

複選題15. For any given 0 ≠ w ∈ , 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.

複選題16. Let A = 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.

複選題17. Let L(.) denote the Laplace transform. Which of the followings are true? (A) L(a1 f1 + a2 ⋅ f2) = a1 ⋅ L(f1) + a2 ⋅ L(f2), for all a1, a2 ∈ 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

複選題18. Consider the differential equation + 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)