Mat 310-f10: Review for Final Exam

1 0 2 0 1 1 (a) Suppose a2 = (1, 2, 2) , a3 = (−2, 0, 1) , a4 = (0, 4, 5) , a5 = (0, 1, 1) . Compute the ranks of A and B. Explain why B can not be obtained from A by a finite number of elementary row operations. Solution: The first 3 columns of B are independent, so its column space has dimension 3, thus rank(B)=3. The second, third and fifth column of A are independent, so its column space has dimension 3, thus rank(A)=3. If B could be obtained from A by elementary row operations, then there would exist an invertible, 3 × 3-matrix C such that Cbi = ai holds for all 1 ≤ i ≤ 6 (bi denotes the i’th column of B). Since the {b2,b3,b4} are idependent, and left multiplication by an invertible matrix C sends an independent set to an independent set, it would follow that {a2, a3, a4} must be independent — which it is not. (b) Suppose that a2 = (1, 1, 1) , a4 = (1, 0, 5) , a6 = (1, 2, 3) ; also suppose that B is obtained from A by a finite number of elementary row operations. Then compute the coordinates of a3. Solutions: We have that ai = Cbi holds for all 1 ≤ i ≤ 6 for some invertible matrix C. Note that b3 = −10b2 + b4 + 2b6. Thus a3 = Cb3 = −10Cb2 + Cb4 + 2Cb6 = −10a2 + a4 + 2a6 = (−7,−6, 1) . Hint: read the proof of Theorem 3.16 on page 190.