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1 - Pivot - Math 1b Practical Basic matrices and solutions...

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Math 1b Practical — Basic matrices and solutions; pivot operations January 3, 2011 We use matrices to model systems of linear equations. For example, the system 2 x 1 3 x 2 5 x 3 7 x 4 + 2 x 5 = 7 x 1 + x 2 5 x 3 + x 4 4 x 5 = 3 x 1 4 x 3 + 3 x 4 + x 5 = 4 (1) has corresponding matrix 2 3 5 7 2 1 1 5 1 4 1 0 4 3 1 7 3 4 . (2) Two matrices (of the same dimensions) are row-equivalent when one can be obtained from the other by a sequence of elementary row operations as described in LADW, Sec- tion 2.1 on page 40. It is important to understand that the systems of linear equations corresponding to two row equivalent matrices will have exactly the same set of solutions. An r by n matrix M with all nonzero rows is basic when its columns include the unit vectors 1 0 0 . . . 0 , 0 1 0 . . . 0 , 0 0 1 . . . 0 , . . . , 0 0 0 . . . 1 (of height r ) (3) in some order. These may be called the special columns of M . A matrix with some rows of all zeros is called basic when the rows of all zeros are at the bottom and the submatrix consisting of the nonzero rows is basic. All matrices in reduced echelon form are basic matrices. Other examples of e.g. 3 × 7 basic matrices are 2 3 0 4 1 0 5 6 7 1 8 0 0 9 5 5 0 5 0 1 5 , 2 1 7 4 1 0 5 6 0 8 8 4 1 9 0 0 0 0 0 0 0 , 2 1 0 4 1 0 0 6 0 1 8 0 0 1 5 0 0 7 0 1 0 .
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