If X₁. X₂, .... are mutually independent norma! random variables, with their means all equal to 3 and variances equal to 10, what is the variance of ?
(A) 400
(D) 200m 
(B) 200
(C) 400m
(E) None ofthe above choices (a)-(d).

Y is distributed N(0,4), and X is distributed N(0,16). Please calculate B(Y2) and E(X3).(A) E(Y2) = 4 and E(X3) = 16(B) E(Y2) = 2 and E(X3) = 64(C) E(Y2) = 4 and E(X3) =0(D) E(Y2) = 2 and E(X3) = 8(E) None of the above choices (a)-(d).

Suppose X and Y have a bivariate normal distribution with covariancc 1. X has a mean of 10 and standard deviation of 4.Y has a mean of 15 and standard deviation of 6. Z = 2X + 3Y. Which of the following statement is false?(A) The mean of Z is 65.(B) 'The variance of Z is 400.(C) The comeiation between X and Y is 1/24.(D) 2 follows a normal distribution.(E) None of the above choices (a)-(d).

 3. X and Z are two independently distributed standard normal random variables. Y = X2+52. Please calculate E(Y). (A)-1.5 (B) 1 (C) 0.5 (D) 2.5

Let A be an m x n matrix whose null space has dimension k. Which conclusion is correct? (A) The dimension of Null(AT) is k. (B) The dimension of row space of A is m-k. (C) The dimension of column space of A is m-k. (D) The dimension of row space of A'is n-k. (E) The dimension of column space of A is n-k.

Let A =xyT, where t and y are two nonzero vectors of Rn, n ＞ 1. Which of the following statements is/are true? (A) rank( A ) = 1 and the range space of A is Spanf{y}. (B) rullity( A ) = 2 and the null space of A is Spanfe,{x，y}. (C) Trace( A ) = 1 and det( A ) =0 (D) A is always diagonalizable. (E) None of the above.

.(5%) Determine the rank of the matrix in the following.  (A)0 (B)1 (C)2 (D)3 (E) 4

11. (10%) Let Q be an n✖ n matrix. Then which of the following set is not a subspace? (A) Col O (B) Null Q (C) rank Q (D) Row Q (E) None of the above

複選題Find a basis of U, if exist, U ={f(x)|f(x)=a+bx+cx2+dx3},p3 =span{1,x,x2,x3}.(10% (A) f(2)=0(B)f(1)=5(C)a+b=c+d, a=2d

複選題Consider the vector space S consisting of all degree-2 polynomials with real coefficients, ie., polynomials in the form of c0 +c1t +c2t2. Define the inner product between two vectors as c2d2. Which of the following statement is/are true? (A). The dimension of S is 2. (B). The polynomials 1 + t, t and 1 + t2 are linearly independent. (C). The set{1 + t, t, 1 + t2} can be a basis for S. (D). The two polynomials 1 - 2t + t2, and -1 + 2t - t2 are orthogonal to cach other. (E). None of the above is true.

(5%) Which of the following is true? (A) Every nonzero subspace of Rn has a unique basis. (B) Every subspace of Rn has a basis composed of standard vectors. (C) The column space of an m✖ n matrix is contained in Rm. (D) If V is a subspace of dimension k, then every generating set for V contains exactly k vectors. (E) The pivot columns of a m ✖ n matrix A form a basis for the column space of A.

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