/*! 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-6.8-7E In Exercises 5-14, the space is ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Q-6.8-7E (page 331)

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

7. Show that \({\left\| {\cos kt} \right\|^2} = \pi \) and \({\left\| {\sin kt} \right\|^2} = \pi \) for \(k > 0\).

Short Answer

Expert verified

It is proved that\({\left\| {\cos kt} \right\|^2} = \pi {\rm{ and }}{\left\| {\sin kt} \right\|^2} = \pi ,\,\,\,\forall k > 0\).

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

Proving the statement

It is given that,\(k \in + {\rm I}\).

Using theinner productrule, we have:

\(\begin{array}{c}{\left\| {\cos kt} \right\|^2} = \int_0^{2\pi } {{{\cos }^2}ktdt} \\ = \frac{1}{2}\int_0^{2\pi } {\left\{ {1 + \cos 2kt} \right\}dt} \\ = \pi \end{array}\)

Similarly,

\(\begin{array}{c}{\left\| {\sin kt} \right\|^2} = \int_0^{2\pi } {{{\sin }^2}ktdt} \\ = \frac{1}{2}\int_0^{2\pi } {\left\{ {1 - \cos 2kt} \right\}dt} \\ = \pi \end{array}\)

Hence, \({\left\| {\cos kt} \right\|^2} = \pi {\rm{ and }}{\left\| {\sin kt} \right\|^2} = \pi ,\,\,\,\forall k > 0\).

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

a. Rewrite the data in Example 1 with new \(x\)-coordinates in mean deviation form. Let \(X\) be the associated design matrix. Why are the columns of \(X\) orthogonal?

b. Write the normal equations for the data in part (a), and solve them to find the least-squares line, \(y = {\beta _0} + {\beta _1}x*\), where \(x* = x - 5.5\).

Let \(\left\{ {{{\bf{v}}_1}, \ldots ,{{\bf{v}}_p}} \right\}\) be an orthonormal set. Verify the following equality by induction, beginning with \(p = 2\). If \({\bf{x}} = {c_1}{{\bf{v}}_1} + \ldots + {c_p}{{\bf{v}}_p}\), then

\({\left\| {\bf{x}} \right\|^2} = {\left| {{c_1}} \right|^2} + {\left| {{c_2}} \right|^2} + \ldots + {\left| {{c_p}} \right|^2}\)

A certain experiment produces the data \(\left( {1,1.8} \right),\left( {2,2.7} \right),\left( {3,3.4} \right),\left( {4,3.8} \right),\left( {5,3.9} \right)\). Describe the model that produces a least-squares fit of these points by a function of the form

\(y = {\beta _1}x + {\beta _2}{x^2}\)

Such a function might arise, for example, as the revenue from the sale of \(x\) units of a product, when the amount offered for sale affects the price to be set for the product.

a. Give the design matrix, the observation vector, and the unknown parameter vector.

b. Find the associated least-squares curve for the data.

Suppose \(A = QR\), where \(Q\) is \(m \times n\) and R is \(n \times n\). Showthat if the columns of \(A\) are linearly independent, then \(R\) mustbe invertible.

In Exercises 7鈥�10, let\[W\]be the subspace spanned by the\[{\bf{u}}\]鈥檚, and write y as the sum of a vector in\[W\]and a vector orthogonal to\[W\].

8.\[y = \left[ {\begin{aligned}{ - 1}\\4\\3\end{aligned}} \right]\],\[{{\bf{u}}_1} = \left[ {\begin{aligned}1\\1\\{\bf{1}}\end{aligned}} \right]\],\[{{\bf{u}}_2} = \left[ {\begin{aligned}{ - 1}\\3\\{ - 2}\end{aligned}} \right]\]
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

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.