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lecture14 - Dierential Equations Linear Algebra Lecture 14...

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Differential Equations & Linear Algebra Lecture 14 Jianli XIE Email: [email protected] Department of Mathematics, SJTU Fall 2010
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Contents Homogeneous systems
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Contents Homogeneous systems Non-singular matrices
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Example Example Solve the system 6 x 3 + 2 x 4 - 4 x 5 - 8 x 6 = 8 3 x 3 + x 4 - 2 x 5 - 4 x 6 = 4 2 x 1 - 3 x 2 + x 3 + 4 x 4 - 7 x 5 + x 6 = 2 6 x 1 - 9 x 2 + 11 x 4 - 19 x 5 + 3 x 6 = 1 . Solution The augmented matrix is A = 0 0 6 2 - 4 - 8 8 0 0 3 1 - 2 - 4 4 2 - 3 1 4 - 7 1 2 6 - 9 0 11 - 19 3 1 which is row equivalent to
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Example(cont.) B = 1 - 3 2 0 11 6 - 19 6 0 1 24 0 0 1 1 3 - 2 3 0 5 3 0 0 0 0 0 1 1 4 0 0 0 0 0 0 0 . The complete solution is x 1 = 1 24 + 3 2 x 2 - 11 6 x 4 + 19 6 x 5 , x 3 = 5 3 - 1 3 x 4 + 2 3 x 5 , x 6 = 1 4 , with x 2 , x 4 , x 5 arbitrary. (Here n = 6 , r = 3 , c 1 = 1 , c 2 = 3 , c 3 = 6; c r < n + 1; r < n. )
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Exercise Exercise For which numbers a and b does the following system have (i) no solution, (ii) a unique solution, (iii) infinitely many solutions. x - 2 y + 3 z = 4 2 x - 3 y + az = 5 3 x - 4 y + 5 z = b. Solution The augmented matrix A is row equivalent to B = 1 - 2 3 4 0 1 a - 6 - 3 0 0 - 2 a + 8 b - 6 .
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Exercise(cont.) Case 1. a = 4 , then B can be reduced to the simplest form 1 0 0 u 0 1 0 v 0 0 1 w , and we have the unique solution. Case 2. a = 4 , then B = 1 - 2 3 4 0 1 - 2 - 3 0 0 0 b - 6 . When b = 6 , we get no solution, whereas if b = 6 , then B is row equivalent to 1 0 - 1 - 2 0 1 - 2 - 3 0 0 0 0 . We obtain the complete solution x = - 2 + z, y = - 3 + 2 z .
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Homogeneous systems A system of homogeneous linear equations is a system of the form a 11 x 1 + a 12 x 2 + · · · + a 1 n x n = 0 a 21 x 1 + a 22 x 2 + · · · + a 2 n x n = 0 . . . a m 1 x 1 + a m 2 x 2 + · · · + a mn x n = 0 . Such a system is always consistent as x 1 = x 2 = · · · = x n = 0 is a solution. This solution is called the trivial solution. Any other solution is called a non-trivial solution.
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Homogeneous systems For example, the homogeneous system x - y = 0 x + y = 0
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