4.Which set is not a vector space?
(A) The set of polynomials of degree exactly 3.
(B) The set of polynomials having'degree 0, 1, 2, or 3, together with the 0-polynomial.
(C) The set of all vectors (x₁, x₂, x₃, x₄) in R⁴ such that x₂ = 5x₃ - 7x₄.
(D) The set of all vectors of the form [a+b³, a-b³, 2a] in R³.
(E) The set of all vectors [a+b³, a-b³] in R².

複選題1. Consider a 3x3 matrix A with non-zero elements. Assume ATAx=0 has non-trivial solutions with two of them being x=[1,1,0]T and [1,0,0]T. Which of the following statements are true for b,y ∈ R3? (A) Ay =b must have no solution for some b and exactly one solution for other b. (B) ATy=[0,0,5]T must have at least one exact solution. (C) The least-squares approximation error for the system ATy =[0,2,2]T is 4. (D) Rank of A =2. (E) A must have zero rows.

複選題2. Suppose  has a reduced row echelon form Which of the following statements are true for x,b ∈ R3? (A) Ax=[3,1,0]T has infinitely many solutions. (B) Ax=[0,1,0]T has no solution. (C) In this case, Column Space of A = Column Space of U. (D) The matrix ATA is invertible. (E) The projection of x=[2,1,11]T onto the Column Space of A is the vector [2,1,0]T .

複選題3. Suppose D=AB with Which of the true? (A) Null Space of D = Null Space of B if and only if a≠5. (B) Column Space of D = Column Space of A if and only if b≠2. (C) There exists values of a, b such that Rank of D=1. (D) Rank of D=2 only if a=5 and b=2. (E) Rank of D=3 only if a≠5 and b≠2.

複選題5. In a vector space V, we have the two sets S = {u1, u2,..., un3 and Q= {V1, V2,..., Vm}. Which of these statements can be true? (A) S is linearly independent, Q spans V, and n ＞ m. (B) V has dimension n, Q spans Y, and m ≥ n. (C) S and Q are bases for V, and n = m. (D) S is linearly independent, Q is a subset of S and Q is linearly dependent. (E) S is linearly dependent, Q is a subset of S and Q is linearly independent.

複選題6. ConsiderEach is obtained from the other by rearranging columns. Which statement is correct? (A) Ker（A）=Ker（B）. (B) Range（A）= Range（B）. (C) A and B have the same reduced row echelon form. (D) Column space of A = Column space of B. (E) Row space of A= Row space of B.

7. Let A be an m x n matrix whose null space has dimension k. Which conclusion is correct? (A) The dimension of Null(AT) is k. (B) The dimension of row space of A is m-k. (C) The dimension of column space of A is m-k. (D) The dimension of row space of A is n-k. (E) The dimension of column space of A is n-k.

8. Let matrices A, B, and C be square matrices. Choose the incorrect arguments (A) The determinant det(AB) = det（A）det（B） (B) Let matrix A can be decomposed into A=QR, where Q is the orthogonal matrix and R is the upper triangular matrix. The determinant det（A） = det(R). (C) Let matrix A be diagonalizable； that is, A=X-1DX. The determinant det（A）=det（D）. (D) Let matrices A and B are similar. The determinant det（A）=det（B）. (E) Let matrix B be the Hermitian transpose of matrix A； that is B=AH. The determinant det（A）=det（B）.

9. Choose the incorrect arguments. (A) Let A be the Hermitian matrix. Then, matrix A is diagonalizable； that is, A=X-1DX. (B) For a square matrix A, the eigenvectors correspond to different eigenvalues are linearly independent. (C) Two similar matrices have the same characteristic polynomial. (D) Let A be an mxn real matrix. Then ATA is diagonalizable. (E) Let matrix A be diagonalizable. Then, matrix A is not singular.

10. Let A be an nxn matrix and b be an arbitrary nx1 vector. Select the incorrect arguments. (A) If determinant det（A）≠ 0, then the system equation Ax=b has exactly one single solution. (B) If matrix A is diagonalizable, then the system equation Ax=b has exactly one single solution. (C) Let matrix A can be decomposed into A=QR, where Q is the orthogonal matrix and R is the upper triangular matrix: The system equation Ax=b has exactly one single solution. (D) Let matrix A have n independent eigenvectors. The matrix A should have n distinct eigenvalues. (E) The system equation Ax=b can be solved by Cramer's rule either the system is consistent or inconsistent.

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