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hw4_solS09 - EE 103 Spring'09 Prof SEJ HW 4 Sol Applied...

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EE 103, Spring ’09, Prof SEJ: HW 4 Sol. Page 1 of 7 Applied Numerical Computing Instructor: Prof. S. E. Jacobsen HW 4 Solution Students: Distributed HW solutions are a component of the course and should be fully understood. Prob 1: 1 n a x Ax a x . Then, 1 1, , 1, , 1 1, , max , , max , , max , , max{ , , }max , , . n j j n j j j j j j n j j j j n j n j j j Ax a x a x a x a x a x a x x x a a x Ax Prob 2: (a) x is the unique solution of Ax b . If the columns of A are linearly dependent, there exists a vector ˆ 0 x so that ˆ 0 Ax . Consider the vector ˆ x x x x . Then ˆ Ax Ax Ax b and the solution x is not unique. (b) Consider the equation Lx b . By forward substitution, it’s clear that the solution is unique. By part (a), the columns of L are linearly independent and L is nonsingular. Or, consider the equation 0 Lx ; by forward substitution, it’s clear that the unique solution is 0 x .
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