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Unformatted text preview: MATH321 HOMEWORK SOLUTIONS HOMEWORK #6 Section 3.1: 1, 2, 3(c), 5 Section 3.2: 1, 2(a)(c)(g), 3, 4(b), 5(a)(f), 6(a)(f), 7, 11, 18 Krzysztof Galicki Problem 3.1.1 (See Answers to Selected Exercises ). Problem 3.1.2 We have A = 1 2 3 1 1 1 1 1 , B = 1 3 1 2 1 1 3 1 , C = 1 3 2 2 1 3 1 . Note that A and B differ only in the second column. In order to get B from A we perform elementary column operation C 2 C 2 2 C 1 on A . That is the same as multiplying by E 1 = 1 2 1 1 from the right, i.e., B = AE 1 . Note that B and C differ only in the second row. In order to get C from B we perform elementary row operation R 2 R 2 R 1 on B . That is the same as multiplying by E 2 = 1 1 1 1 from the left, i.e., E 2 B = C . We can transform C into I 3 by the following sequence of elementary row operations: R 2 ( 1 / 2) R 2 gives C = 1 3 1 1 1 3 1 , R 3 R 3 R 1 gives C = 1 3 1 1 3 2 , R 3 R 3 + 3 R 2 gives C = 1 3 1 1 1 , R 2 R 2 R 3 gives C = 1 3 1 1 , R 1 R 1 3 R 3 gives C = 1 1 1 . Problem 3.1.3 (c) E 1 = 1 1 2 1 . Problem 3.1.5 Note that the rows of A are the columns of A t and vice versa. Hence A is elementary (obtained from I n by an elementary row operation of some kind) if and only if A t is elementary (obtained by an elementary column operation of the same kind).is elementary (obtained by an elementary column operation of the same kind)....
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This note was uploaded on 11/30/2009 for the course MATH 115A taught by Professor Liu during the Winter '07 term at UCLA.
 Winter '07
 Liu
 Math, Linear Algebra, Algebra

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