複選題

23. Let A ∈ and x ∈ . Which of the following statements are true?
(A) If A is singular, then O is an eigenvalue of A.
(B) A and share the same eigenvalues and eigenvectors.
(C) If A is diagonalizable, then A has n distinct eigenvalues.
(D) Suppose that A is nonsingular. The condition Ax = Ix implies .

(E) Supposc that all the eigenvalues of A are real and positive. Then we have ＞ 0.

複選題19. Consider the linear equation Ax = b with A ∈ . Which of the following statements are true?(A) If rank = m, then there exists at least one solution.(B) Ifrank = n, then there exists exactly one solution.(C) Ifrank = n, then the column vectors of A are linearly independent.(D) Ifn ＞ m, then there exists at least one solution.(E) Ifm ＞n, then there exists at most one solution.

複選題20. Consider the linear mapping L: V → W. Let be the zero vectors in V and W, respectively.Which of the following statements are true?(A) The condition L(V1) = L(V2) impliesV1 = V2.(B) For any W ∈ W, there exists V ∈ V such that L(V) = W.(C) If L is one-to-one, then L(v) = 0w implies v = Ov.(D) If V1,V2,..., are linearly independent, L(V1), L(V2),.., are also linearly independent.(E) The condition C1V1+C2V2+..+=implies c1L(V1)+c2L(V2)⋯+=.

複選題21. Given vectors x, y,z in and matrices A, B, C in . Which of the following statements aretrue?(A) (B) (C) (A+B)(A-B) = A2 - B2(D) If AC = BC and C is not the zero matrix, then A = B.(E) If AB equals the zero matrix, then BA also equals the zero matrix.

複選題22.Let A ∈ , denote the column space of A, denote the null space of A,and dim(S) denote the dimension of a subspace S. Which of the following statements are true?(A) For any x ∈ , there exists u ∈ and v ∈ such that x = u + v.(B) Suppose u ∈ and v ∈ . Then = 0.(C) dim+ dim =n(D) For any y ∈ , there exists x ∈ such that y = .(E) Ify ∈ , then y is the zero vector in .

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