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

12. (10%) Let M be a subset of R3 containing two or more vectors. Then: (A) The span of any two vectors in M is a plane in R3 (B) Every vector in M is in the span of M (C) If M contains more than threc vectors, then M is linearly independent (D) The span of any two nonzero vectors in M is a plane in R3 (E) None of the preceding statements is true

(5%) Let For what valuc(s) of h is b in the planc spanned by al and az? (A) 0 (B) -10 (C) 16 (D) -17 (E) 20

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.

(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.

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.

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

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

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