複選題16. (5%) Which of the following statements are correct? (A) If λ is an eigenvalue of an n x n matrix A, then for any (B) If there exists some such that Av =λv, then λ is an eigenvalue of A. (C) If two matrices have the same characteristic polynomials, then they have the same eigen- values. (D) A diagonal matrix is always diagonalizable. (E) Let A and P be n X n matrices. If the columins of P form a set of n linearly independent eigenvectors of A, then AP is a diagonal matrix.

複選題15. (5%) Which of the following statements are correct? (A) A basis for a subspace is a linearly independent subset of the subspace that is as large as possible. (B) If V is a subspace of dimension k, then every generating set for V contains exactly k vectors. (C) is any basis for Rn, then there exists a vector v so that v can not be expressed as a linear combination of bi, ba...., bn. (D) Let T be a linear operator on Rn, and u and V be two bases forRn. Denote [T]Bas the representative matrix of T with respect to any basis B for Rn. Then, [T]u and [T]v are similar. (E) If B is a basis for Rn and T is the identity operator on R", then [T]B = In.

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