110年 - 110 國立臺灣大學_碩士班招生考試_電信工程研究所丙組:工程數學-線性代數#103391

11. (10%) Consider a matrix
Which of the following is an eigenvalue of this matrix M? (A)1 (B)-3 (C)9 (D) -1+2j (E) -1+4j

12. (10%) Consider the matrix M in (t). It is known that M has four distinct eigenvalues, and let us denote the four corresponding cigenvectors as Which of the following mattices also has as its eigenvectors? (A) (B)  (C) (D)(E)

13. (10%) Consider the subspace S = span( B ) spanned by the following elements in the vector space :

 Which of the following does not belong to S?(A)(B)(C)(D)(E)

14. (10%) Let S = span( B ) be a subspace of the vector space and B is defined as in (t). Consider the following mappings: Choose the correct statement(s) from the following. (A) Both T, and Tz are linear transforms from S to. (B) Ts is a linear transform from S to . (C) The range of T₁is a subspace of  and its dimension is 2. (D) The range of T₂ is a subspace of  and its dimension is 2. (E) The range of T₃ is a subspace of and its dimension is 2.

15. (10%) For _n > 1, let S_n denote the set of all symmetric matrices and T_n denote the set of all skew-symmetric matrices, that is, T_n = {A | A is a n x n matrix such that A^T = -A}. Let U be a n x n matrix such that U^TU = I_n. Furthermore, let
 Wu ={M I M is a n x n matrix such that U^TMU is a diagonal matrix}.
For a n xn matrix A, let Au denote the unique matrix B in Wu such that is minimized. Choose the correct statement(s) from the following.
 (A) S_n is a subspace of and its dimension is n. (B) T. is a subspace of  and its dimension is n. (C) W_U is a subspace of and its dimension is n. (D)(E)

1. (5%) Let A and B be two n x n syinmetric matrices, n ≥ 1.
Then, rank(AB) = rank(BA).

4. (5%) Let A be a n x n invertible matrix, n≥ 1. Let u be an eigenvector of A. Then, u is also an eigenvector of .

5. (5%) Let A, B be n x n matrices, n ≥ 1, andλ be an eigenvalue of the matrix product AB. Then, λis also an eigenvalue of the matrix product BA.

6. (5%) Let A be a n x n symmetric matrix, n ≥ 1, and λ be an eigenvalue of A. Then, A is a real number.

7. (5%) Let A be a n x n skew-symmetric matrix, n ≥ 1, that is, A^T = -A. Then, det(A)≥0.

8. (5%) Consider a system of linear equations At = b, where A is a 2 x 3 matrix, I is a 3 x 1 column vector, and b is a 2 X 1 column vector. LetIn other words, all entries of A are non-zero. Then, it is impossible to uniquely determine t1 from the above system of linear equations.

9. (5%) Let S₁ and S₂ be subspaces ofRⁿ and S = S₁+ S₂, the sum of the two subspaces. Let D₁ = , the set of elements in S but not in S₂.
Then, D₁ is also a subspace, and its dimension dim(D₁) = dim(S₁).

10. (5%) Let S be a subspace of Rⁿ, n ＞ 1. Let dim(S) = m, 1 ＜m ＜n. Let A be a n xn diagonal matrix with positive diagonal entries, but not all entries are equal. Consider the following set:

 Then, T is also a subspace, and its dimension dim(T) = n - m.