(10%)There are six vectors shown below. Please determine the right description
(A) are linear independent
(B) are linear independent
(C) are linear independent
(D) are linear independent 

複選題 (5%) Which of the following lists of vectors in R3 are linearly dependent? (A)u=(1,2,5),t(2,5,1),u元(5,2,1) (B)u=(1,2,5), =(1,5,6), w=(0, 0, 0) (C)น=(2,-4,8),=(3 ,-6, 12),w=(1,2,6) (D)u-(1,2,3),=(4,5,6),w=(1,4,6),x-(3,4,6)

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.

(5%) Find the dimension of the subspacc (A) 2 (B) 3 (C) 4 (D) 1 (E) 0 

複選題Which of the following sets is basis (are bases) for R3? (A){10,1,0]T,[0,0,1]T,[1,0,0T}. (B){-2,4,-6]T,[1,-2,3]T}. (C){|0.14,0, -0.1]T, [-1,-0.2,0.4]T,[0.5,0.5,-1)T} (D) {1-1,2,3]T, [8,-7,6]T, [4,-2,3]T,[9,0,5]T}, (E) None of the above are true.

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