3 - Equivalence and Inverses

3 - Equivalence and Inverses - Math 1b Row-equivalence;...

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Math 1b — Row-equivalence; matrix inverses January 7, 2011 Recall that matrices A and B are row-equivalent when one can be obtained from the other by a sequence of elementary row operations. An elementary row operation on a matrix M g ivesusamatr ixwhoserows M 0 whose rows are linear combinations of the rows of M . Since elementary row operations can be ‘undone’ by other elementary row operations, the rows of M are linear combinations of the rows of M 0 . It follows that if A and B are row-equivalent, then the rows of A are linear combinations of the rows of B ,andtherowso f B are linear combinations of the rows of A . (Later, we will say that A and B have the same row space .) To review one item from the handout on matrix multiplication, recall that given matrices A and B , the equation A = CB holds for some matrix C if and only if the rows of A are linear combinations of the rows of B . For example, ± a 11 a 12 a 13 a 21 a 22 a 23 ² = ± 2 345 67 89 ² b 11 b 12 b 13 b 21 b 22 b 23 b 31 b 32 b 33 b 41 b 42 b
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This note was uploaded on 11/10/2011 for the course MA 1B taught by Professor Aschbacher during the Winter '08 term at Caltech.

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3 - Equivalence and Inverses - Math 1b Row-equivalence;...

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