2. Show that the scalar product of two vectors remains invariant at linear transformations of the vectors if and only if the transformations are orthogonal. That is, given two vectors a; and bi, and a simultaneous linear transformation ai + ái = Rijaj bi → bį = Rijbj (5) show that we have aźb; = a;b; if an only if the matrix R is orthogonal. The Einstein summation convention is implied throughout. Hint: If you have more than one implicit summation in the same formula, make sure to prop- erly differentiate between the summations by denoting the corresponding summation indices with different letters! In other words, an index cannot be repeated more than once in any given formula.

2. Show that the scalar product of two vectors remains invariant at linear transformations of the vectors if and only if the transformations are orthogonal. That is, given two vectors a; and bi, and a simultaneous linear transformation ai + ái = Rijaj bi → bį = Rijbj (5) show that we have aźb; = a;b; if an only if the matrix R is orthogonal. The Einstein summation convention is implied throughout. Hint: If you have more than one implicit summation in the same formula, make sure to prop- erly differentiate between the summations by denoting the corresponding summation indices with different letters! In other words, an index cannot be repeated more than once in any given formula.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter7: Distance And Approximation

Section7.4: The Singular Value Decomposition

Problem 27EQ

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