5、Let V be the vector space of all A ∈ R^m✖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,..., B_n} be a basis for V. Suppose U is a linear transformation from V to Rn given by U(c₁B₁ + c₂B₂ + ... + c_nB_n) =. Which of the following statements is/are true?
(A) Let {A₁, A₂,..., A_k} be linearly independent over R. The set {U(A₁), U(A₂),…, U(A_k)} 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(B_k)} 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.

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.

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.

11、Consider the first order differential equation xy'(x) = with initial condition y(2) = 0. Which of the following statements is/are true?(A) It is a linear differential equation for the dependent variable y. (B) y(x) is a parabolic function of x. (C) y'(x) = -x. (D) y"(x) = -. (E) None of the above is true.

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