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# lecture_27 - MA 265 LECTURE NOTES WEDNESDAY MARCH 19...

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MA 265 LECTURE NOTES: WEDNESDAY, MARCH 19 Homogeneous Systems Relationship with Nonhomogeneous Systems. In general, consider a nonhomogeneous system A x = b . Recall that the set of solutions does not form a linear space: x R n A x = b . Say that x p is a particular solution, and let x be any solution. Denote x h = x - x p . Then we find that A x h = ( A x ) - ( A x p ) = b - b = 0 . Hence x h is a solution to the homogeneous equation. We summarize this as follows: Any nonhomogeneous system A x = b has the general solution x = x p + x h where x p is one solution to the nonhomogeneous system and x h is the general solution to the homogeneous system. In other words, x R n A x = b = x p + x h x h W = x p + W where W = x R n A x = 0 . Here W is a linear space; it is the null space of A . Similarly, x p + W is the translate of the linear space W . Example. Consider the following linear system: x + 2 y - 3 z = 2 2 x + 4 y - 6 z = 4 3 x + 6 y - 9 z = 6 We can express this in the form A x = b in terms of the matrices A = 1 2 - 3 2 4 - 6 3 6 - 9 , x = x y z , and b = 2 4 6 . The augmented matrix for this system is A b = 1 2 - 3 2 2 4 - 6 4 3 6 - 9 6 Performing Gauss-Jordan Reduction gives the reduced row echelon form 1 2 - 3 2 0 0 0 0 0 0 0 0 = x + 2 y - 3 z = 2 We see that y = r and z = s are arbitrary variables. Hence the general solution is

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lecture_27 - MA 265 LECTURE NOTES WEDNESDAY MARCH 19...

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