/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Q-6.8-6E 6. In Exercises 5-14, the space ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Q-6.8-6E (page 202)

6. In Exercises 5-14, the space is \(C\left[ {0,2\pi } \right]\) with inner product (6).

6. Show that \(\sin mt{\rm{ and cos}}nt\) are orthogonal for all positive integers \(m{\rm{ and }}n\).

Short Answer

Expert verified

The functions \(\sin mt{\rm{ and cos}}nt\) are orthogonal.

Step by step solution

01

Inner Product 

The Inner Productfor any two arbitrary functions is given by:

\(\left\langle {f,g} \right\rangle = \int_0^{2\pi } {f\left( t \right)g\left( t \right)dt} \)

02

Prove the statement

It is given that,\(\left( {m,n} \right) \in + {\rm I}\).

Using theinner productrule, we have:

\(\begin{array}{c}\left\langle {\sin mt,\cos nt} \right\rangle = \int_0^{2\pi } {\sin mt\cos ntdt} \\ = \frac{1}{2}\int_0^{2\pi } {\left\{ {\sin \left( {m + n} \right)t + \sin \left( {m - n} \right)t} \right\}dt} \\ = 0\end{array}\)

Hence,\(\sin mt{\rm{ and cos}}nt\)are orthogonal.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the length of each of the vectorsv鈬赌In exercises 1 through 3.

  1. v鈬赌=[711]

Consider the vectors

ur=[11...1]andvr=[11...1]inRn

a.For n =2,3,4 , find the angle胃between role="math" localid="1659432008110" u鈯�and v鈯�. For n=2and 3, represent the vectors graphically.

b.Find the limit of 胃as napproaches infinity.

Consider an invertible n脳nmatrix A. Can you write Aas A=LQ, where Lis a lowertriangular matrix andQis orthogonal? Hint: Consider the QRfactorizationof AT.

Using paper and pencil, perform the Gram-Schmidt process on the sequences of vectors given in Exercises 1 through 14.

11.[2306],[44213]

Leg traction.The accompanying figure shows how a leg may be stretched by a pulley line for therapeutic purposes. We denote by F1the vertical force of the weight. The string of the pulley line has the same tension everywhere. Hence, the forces role="math" localid="1659529616162" F2and F3have the same magnitude as F1. Assume that the magnitude of each force is 10 pounds. Find the angle 胃 so that the magnitude of the force exerted on the leg is 16 pounds. Round your answer to the nearest degree. (Adapted from E. Batschelet, Introduction toMathematics for Life Scientists, Springer, 1979.)
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.