20、Consider the following partial differential equation for the bivariate function v(x, t)

subject to conditions

Express the solution v(x, t) in the following form

for some constants an, bn, pn, qn, rn ∈ R satisfying 0 ≤ r₀ ＜ r₁ ＜ r₂ ＜ ... and ＞0
for n = 1, 2, ... Which of the following statements is/are true?
(A) a₀ = -
(B) b₁ =
(C) p₃ =
(D) r₄ =
(E) None of the above is true.

複選題1. Letbe the reduced row echelon form of a matrix A = [a1 a2 a3 a4 a5]. Which of the following statements is/are true? (A) a2 is the 4 × 1 zero vector. (B) The three column vectors a1, a4 and a5 are linearly independent. (C) The column space of R is the same as the column space of A. (D) Let B = [a1 a2 a3] be a submatrix of A. The reduced row echelon form of B is not always a submatrix of R. (E) None of the above is true.

2、Continue from Question. Which of the following statements is/are true?(A) The nonzero rows of R form a basis for col(AT). (B) Let C be the reduced row echelon form of RT. Then CTC is the identity matrix. (C) Let E be a 3 × 5 matrix and be the reduced row echelon of E. We can obtain the intersection of the right null space of A and the right null space of E from R and D without knowing A and E. (D) Let Q be a 5 x 5 matrix and rank(Q) = 3. Then rank(AQ) = 3. (E) None of the above is true.

複選題3、Let U and V be subspaces of Rn and T be a linear transformation from Rn to Rm. Which of the following sets is/are subspace(s) of Rn?(A) {a + b ∈ Rn : a ∈ U and b ∈ V}. (B) {a ∈ Rn : a ∈ U or a ∈ V}. (C) {a ∈ Rn : a ∈ U and a ∈ V}. (D) {a ∈ Rn : T(a) = 0}. (E) None of the above is true.

複選題4、Let A, B, and C be n×n matrices for some positive integer n. Which of the following statements is/are true? (A) det( A ) = 0 implies det(R) = 0, where R is the reduced row echelon form of A. (B) If AB = CA = In, then B = C. (C) If det( AB ) ≠ 0, then A is invertible. (D) rank( AB )=rank( BA ). (E) None of the above is true.

5、Let V be the vector space of all A ∈ Rm✖R, with the operations of matrix addition and multiplication of a matrix by a real scalar. Let T be a linear transformation from V to V, and {B1, B2,..., Bn} be a basis for V. Suppose U is a linear transformation from V to Rn given by U(c1B1 + c2B2 + ... + cnBn) =. Which of the following statements is/are true?(A) Let {A1, A2,..., Ak} be linearly independent over R. The set {U(A1), U(A2),…, U(Ak)} can be linearly dependent over R. (B) Suppose the dimension of the range space of T is k over R. Then {T(B1), T(B2),……, T(Bk)} are linearly independent over R. (C) n = m. (D) Let c = U(B1) and a = U(T(B1), then aTa = cTc. (E) None of the above is true.

6、For the matrix which of the following statements is/are true? (A){rank}( A ) = 3. (B) The sum of eigenvalues of Ais 6. (C) A is similar to B = (D) The system of linear equations Ax = [-1 1 1]T \end{bmatrix} has a solution. (E) None of the above is true.

複選題7、For a non-zero matrix A∈Rm✕n, define the maximum Frobenius norm||Ax|| over all unit-norm vectors x. Which of the following statements is/are true? (A)σ2 is the maximal eigenvalue of AAT. (B) If x0 is an optimal solution to equation (1), i.e., ||Ax0||= σ, then there exists a vector x1 such that Ax1 = 0 and x1Tx0≠0. (C) Assume m = n and A is nonsingular. Then. (D) Assume m = n. Then In + A is singular only if σ≥ 1. (E) None of the above is true.

複選題8、Let V together with the inner product (•, •) v be an inner product space over R. For acollection of N linearly independent vectors {w1, ..., wN} in V, let M = [mi，,j] ∈RN✖ Nbethe corresponding Gram matrix, i.e., the (i, j)-th entry of M is given by m{i，,j} = (wi, wj)V.Define the following set of real-valued functions f: V→Rwhich is a vector space over R when combined with standard addition and scalar multiplica-tion of real-valued functions. For anyin the vector space F, defineWhich of the following statements is/are true?(A) M is positive definite.(B) If m1,1 = 2, m2,2 = 1, and m1,2 = -, then ||w1 - w2||V = 4, where ||\cdot||_V is the vector norm induced by the inner product(•, •) v.(C) (f, g)F is an inner product for elements f, g∈ F.(D) Given g(y) = (v1, y)V, we have (f, g)F = f(w1) for all f∈ F.(E) None of the above is true.

複選題9、Which of the following statements is/are true?(A) Let x and y be two non-zero vectors in Rᵀ. The matrix xyᵀ is diagonalizable ifyᵀx ≠ 0.(B) If A ∈ Rⁿˣⁿ is symmetric and positive definite, then A + A⁻¹ - 2Iₙ is positive semi- definite.(C) For u, v ∈ Rⁿ such that uᵀv ≠ 0, Rⁿ = (span{u})⊥⊕span{u}, where ⊕ is theoperation of direct sum of two vector spaces.(D) For the real matrix Q = Iₙ - uuᵀ, where u ∈ Rⁿ, we have det(Q) + rank(Q) = n.(E) None of the above is true.

複選題10、Let P ∈ Rⁿˣⁿ be the orthogonal projection matrix, with respect to the Euclidean inner product, of vectors in Rⁿ onto col( A ) for some nonzero matrix A ∈ Rⁿˣᵏ. Which of the following statements is/are true?(A) tr(PᵀP) ≥ rank(P).(B) P can have an eigenvalue larger than 2.(C) P²x ≠x for all x ∈ Rⁿ.(D) If k ＜ n and b ∈ Rⁿ, the system of linear equations Ax = b has a unique least squares solution.(E) None of the above is true.

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