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hw1 - Homework 1 Foundations of Computational Math 1 Fall...

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Homework 1 Foundations of Computational Math 1 Fall 2010 The solutions will be posted on Wednesday, 9/8/09 Problem 1.1 This problem considers three basic vector norms: . 1 , . 2 , . . 1.1.a . Prove that . 1 is a vector norm. 1.1.b . Prove that . is a vector norm. 1.1.c. Consider . 2 . (i) Show that . 2 is definite. (ii) Show that . 2 is homogeneous. (iii) Show that for . 2 the triangle inequality follows from the Cauchy inequality | x H y | ≤ x 2 y 2 . (iv) Assume you have two vectors x and y such that x 2 = y 2 = 1 and x H y = | x H y | , prove the Cauchy inequality holds for x and y . (v) Assume you have two arbitrary vectors ˜ x and ˜ y . Show that there exists x and y that satisfy the conditions of part (iv ) and ˜ x = αx and ˜ y = βy where α and β are scalars. (vi) Show the Cauchy inequality holds for two arbitrary vectors ˜ x and ˜ y . Problem 1.2 What is the unit ball in R 2 for each of the vector norms: . 1 , . 2 , . ? Problem 1.3 Consider the matrices B 1 = 1 1 1 1 0 1 B 2 = 0 2 0 2 - 1 1 1.3.a . Show that they have the same range space.
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