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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|| = ||*||?|

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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||≤|||| (this is called the Bessel's inequality). When do we have ||UT||||||?
Transcribed Image Text: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||≤|||| (this is called the Bessel's inequality). When do we have ||UT||||||?
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