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Numerical Methods Proofs

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a) Show that the l_inf vector norm satisfies the three properties
i. ||x|| > 0 if x =/ 0 and x belongs to R^n
ii. ||lambda(x)|| |lambda| ||x|| if lambda belongs to R and x belongs to R^n
iii. ||x + y|| <= ||x|| + ||y|| for x, y belonging to R^n.

b) consider the matrix
4 2 1
A = 2 5 -2
1 -2 7

Compute ||A||_infinity and find a vector x such that ||A||_infinity = ||A_x|| / ||x||_infinity.

c) Compute the condition number, k(A), for the above matrix A.

d) Prove that is ||A|| < 1, then

||(I - A)^1|| >= 1/(1+||A||).

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Solution Summary

This solution shows how to complete a proof regarding properties of a vector norm, compute a condition number, compute ||A||sub infinity, and complete a proof involving ||A||. This solution is provided in an accompanying pdf file.

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