7. (5%) Consider a regular transition matrix for a three-state Markov chain, with the 10.3 initial probability vector Find the proportions of objects eventually.

3. (5%) Let B=[(1,1),(2,1)3 be an ordered basis of R2. Suppose P is the orthogonal projection onto spanf(0,1)]. What is the matrix representation [P]β of P in B?

4. (5%) Which statement below is correct? (A) Any determinant is a linear transform (B) Similar matrices always have the same eigenvalues (C) The zero vector space has no basis (D) Every system of n linear equations in n unknowns can be solved by Cramer's rule (E) Only square matrices have a conjugate-transpose J112

5. (5%) Which of the following number can make the matrix B positive definite ? (A)-1 (B) 0 (C) 1 (D) 2 (E) i U112 B

6. (5%) Determine (A) 1 (B)-1 (C) 0 (D) i (E) Does not exist

複選題8. (5%) Which of the following vectors are linearly independent in the vector space R3? (A)[1,1,3]T,[0,2,1]T,[2,0,5]T (B)[1,1,0]T,[0,1,1]T,[2,1,-1]T (C)[4,1,-3]T,[1,0,-1]T,[1,2,1]T (D)[1,-1,2]T,[1,1,7]T,[2,0,-1]T (E)[1,-1,2]T,[1,0,7]T,[2,-3,-1]T

9. (5%) Consider the system of linear equations: x +2y +4z =8 x+y+3z=-2 x-3y-z=0 Which of the following is NOT the least square solution of this system? (A) [2,1,0]T (B) [-4,-2,3]T (C) [4,2,-1]T (D) [-2,-1,1]T (E) [-2,-1,2]T 

10. (5%) Which of the following matrices is diagonalizable?

11. (5%) Suppose that the characteristic polynomial of a matrixA is p(λ)=λ(λ-1)(λ-2)2(λ+3)3. Which of the following statements about the matrix A is correct? (A) The size of A is 4 X 4 (B) A is invertible (C) det( A ) =-128 (D) (2I -A)x= 0 has nontrivial solutions (E) A has nullity O

12. (5%) Let A be an n by n matrix. Which of the following statement is NOT identical to others? (A) Ax = 0 has a nontrivial solution (B) The rank of A is less than n (C) The dimension of A's null space is positive (D) A is not diagonalizable (E) The singular values of A are not all positive 乙L

13. (5%) Let vi, ... , Vn be vectors in a vector space V for n ＞ 0.Which statement is NO'T correct? (A) If the dimension of V is n, and vi, ... , In span V, then yi, ... , vn are linearly independent (B) If the dimension of V is m, and m ＞ n, then vi, ... , vn can be extended to form a basis for V (C) If the dimension of V is m, and m ＜ n, then vi, ... , In are linearly dependent (D) If both fu, ... , Ms and fu, ... , um； are bases for a vector space V, then n = m (E) A vector v in Span(vI, v) can be written uniquely as a linear combination of VI, ... , In if and only if v, , vn are linearly independent [4 7 37