Parts C and D 1. Show that the following statements on orthonormal vectors and orthogonal matrices are true.(a) If U € Rnxn and V € Rnxn are orthogonal, then so is UV.(b) If U E Rxn is orthogonal, then so is U-1(c) If U € Rmxn satisfies UTU = In and UUT = Im, then m = n.(d) If the columns of UE R* are orthonormal, then for any ze R", we have |UT||≤|||| (thisis called the Bessel's inequality). When do we have ||UT||||||?

Solution of the question is given below

1. Show that the following statements on orthonormal vectors and orthogonal matrices are true. (a) If U € Rnx and V R* are orthogonal, then so is UV. (b) If U € Rnxn is orthogonal, then so is U-¹. (c) If U € Rmx satisfies UTU = In and UUT = Im, then m = n. (d) If the columns of UE R* are orthonormal, then for any ze R", we have |UT||≤|||| (this is called the Bessel's inequality). When do we have ||UT|| = ||*||?|

1. Show that the following statements on orthonormal vectors and orthogonal matrices are true. (a) If U € Rnx and V R* are orthogonal, then so is UV. (b) If U € Rnxn is orthogonal, then so is U-¹. (c) If U € Rmx satisfies UTU = In and UUT = Im, then m = n. (d) If the columns of UE R* are orthonormal, then for any ze R", we have |UT||≤|||| (this is called the Bessel's inequality). When do we have ||UT|| = ||*||?|

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter5: Orthogonality

Section5.3: The Gram-schmidt Process And The Qr Factorization

Problem 10AEXP

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