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lecture_8- Echelon Form of a Matrix

# lecture_8- Echelon Form of a Matrix - MA 265 LECTURE NOTES...

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Unformatted text preview: MA 265 LECTURE NOTES: FRIDAY, JANUARY 25 Echelon Form of a Matrix Elementary Row Operations. Let A = [ a ij ] be an m × n matrix. Denote its i th row as r i = bracketleftbig a i 1 a i 2 ··· a in bracketrightbig where A = a 11 a 12 ··· a 1 n . . . . . . . . . . . . a i 1 a i 2 ··· a in . . . . . . . . . . . . a m 1 a m 2 ··· a mn = r 1 . . . r i . . . r m . We define an elementary row operation as any of one of the following three types of operations: • Type I: Interchange any two rows. For example, if we interchange row i and row j , we denote this by “ r i ↔ r j ”. • Type II: Multiply a row by a nonzero number. For example, if we multiply the i th row by a number k , we denote this by “ k r i → r i ”. • Type III: Add a multiple of one row to another. For example, if we add k times the i th row to the j th row, we denote this by “ k r i + r j → r j ”. We say that two m × n matrices A and B are row equivalent if B can be produced by applying a finite sequence of elementary row operations to A ....
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lecture_8- Echelon Form of a Matrix - MA 265 LECTURE NOTES...

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