hw4_solS09

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

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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 ax Ax    . Then,  1 1, , , 11 , , max , , max , , max , , m a x { ,, } m a x ,, . n jj n n nj n j Ax a x a x x xa a x A x         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 xxx  . 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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EE 103, Spring ’09, Prof SEJ: HW 4 Sol. Page 2 of 7 (c) (i) To derive the inverse of n L , we solve the equation 1,1 11 01 0 0 tt t n nn l ae I cL I fD     for , , , t ae f D . Using matrix multiplication, derive the system of equations to be solved and verify that the solution is 1 1 11 1 1 0 t l lLc L (ii) By induction: Clearly a 2x2 nonsingular lower triangular matrix has a lower triangular inverse (show this). Therefore, assume the result is true for any (1 ) x ) nonsingular lower triangular matrix; to show the result is true for any x nonsingular lower triangular matrix. The latter follows immediately from the above since the induction assumption implies 1 1 n L is lower triangular.
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This note was uploaded on 06/01/2010 for the course EE EE 103 taught by Professor Jacobsen during the Spring '09 term at UCLA.

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

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