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

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

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

14.(5%) Which one is NOT an eigenvalue of (A)1 (B) 2 (C) 3 (D) 4 (E) 5 O I 4 -2]

15.(5%) Let  Which value of a can make A defective?  (A) No value of a can make A defective (B) -2 (C) 2 (D) 0 6 1 10 (E) 1

16. (5%) Kim is at a train station, waiting to make a phone call. Two public telephone booths, next to each other, are occupied by two callers, and 1 1 persons are waiting in a single line ahead of Kim to call. If the duration of each telephone call is an independent exponential random variable with λ= 1/3, what is the variance of Kim's waiting time?Kim's waiting time is defined as the time interval from now to the moment that a phone is available for Kim to make his call. (A) 18 (B) 27 (C) 99/4 (D) 99 (E) 108

17. (5%) Let X and Y be independent random points from the interval (-1,1). Find E(max(X,Y)). (A) 2/3 (B) 1/2 (C) 1/3 (D) 1/4 (E)0

18.(5%) A stick of length 1 is broken into two pieces at a random point. Find the correlation coefficient and the covariance of these pieces. (A)1/2 (B) -1/2 (C) 1/12 (D)-1/12 (E) 0