/*! 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} Q6E If A聽is an聽n脳mmatrix, is the ... [FREE SOLUTION] | 91影视

91影视

Find study content
Learning Materials

Discover learning materials by subject, university or textbook.

Explanations

Textbooks
All Subjects

Anthropology

Archaeology

Architecture

Art and Design

Bengali

Biology

Business Studies

Greek

Chemistry

Chinese

Combined Science

Computer Science

Economics

Engineering

English

English Literature

Environmental Science

French

Geography

German

History

Hospitality and Tourism

Human Geography

Italian

Japanese

Law

Macroeconomics

Marketing

Math

Media Studies

Medicine

Microeconomics

Music

Nursing

Nutrition and Food Science

Physics

Polish

Politics

Psychology

Religious Studies

Sociology

Spanish

Sports Sciences

Translation

All Subjects

Biology

Business Studies

Chemistry

Combined Science

Computer Science

Economics

English

Environmental Science

Geography

History

Math

Physics

Psychology

Sociology
Features
Features

Discover all of these amazing features with a free account.

Flashcards

91影视 AI

Notes

Study Plans

Study Sets
What鈥檚 new?

Flashcards
Study your flashcards with three learning modes.

Study Sets
All of your learning materials stored in one place.

Notes
Create and edit notes or documents.

Study Plans
Organise your studies and prepare for exams.
91影视
Discover

All the hacks around your studies and career - in one place.

Find a degree

Magazine
Featured

Magazine
Trusted advice for anyone who wants to ace their studies & career.

The largest student job board with the most exciting opportunities.

Verified student deals from top brands.

Discover our mobile app to take your studies anywhere.

Learning Materials

Features

Discover

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 5: Q6E (page 245)

If Ais an $n 脳 m$ matrix, is the formula $im (A) = im ({AA}^{T})$ necessarily true? Explain.

Short Answer

Expert verified

Yes.

Step by step solution

01

Determine the image property.

Consider a $n 脳 m$ matrix A.

Theorem: For any matrix B, $im (B) = {kerB}^{鈯�}$ .

By the theorem, $i m (A) = {(k e r A)}^{鈯�}$ .

Theorem: If B is an $n 脳 m$ matrix then, $\ker (B) = \ker (B^{T} B)$ .

By the theorem, $k e r (A) = k e r (A^{T} A)$ .

Take $鈯�$ both side in the equation $k e r (A) = k e r (A^{T} A)$ as follows.

$k e r (A) = k e r (A^{T} A) {(k e r (A))}^{鈯�} = {(k e r (A^{T} A))}^{鈯�}$

Theorem: For a subset V of ${鈩�}^{n}$ then ${(V^{鈯�})}^{鈯�} = V$ .

By the theorem,localid="1659501115571" ${(k e r (A^{T} A))}^{鈯�} = i m (A^{T} A)$ .

Compare the equations ${(k e r (A^{T} A))}^{鈯�} = i m (A^{T} A), {(k e r (A))}^{鈯�} = {(k e r (A^{T} A))}^{鈯�} a n d i m (A) = {(k e r A)}^{鈯�}$ ,and as follows.

$i m (A^{T} A) = {(k e r (A^{T} A))}^{鈯�} = {(k e r (A))}^{鈯�} i m (A^{T} A) = i m (A)$

Hence, for any $n 脳 m$ matrix A the equation $i m (A^{T} A) = i m (A)$ holds true.

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) Consider an matrix A such that $A^{螕} A = I_{m}$ . It is necessarily true that? Explain.

(b) Consider an $n 脳 n$ matrix A such that $A^{T} A = I_{n}$ . Is it necessarily true that $A A^{T} = I_{n}$ ? Explain.

Use the various characterizations of orthogonal transformations and orthogonal matrices. Find the matrix of an orthogonal projection. Use the properties of the transpose. Which of the matrices in Exercise 1 through 4 are orthogonal? $[\begin{matrix} 0.6 & 0.8 \\ 0.8 & 0.6 \end{matrix}]$ .

Find the length of each of the vectors $\overset{鈬赌}{v}$ In exercises 1 through 3.

$\overset{鈬赌}{v} = [\begin{matrix} 7 \\ 11 \end{matrix}]$

Consider a subspace Vof $R^{n}$ and a vector $\overset{鈫�}{x}$ in $R^{n}$ . Let $\overset{鈫�}{y} = p r o j_{V} \overset{鈫�}{x}$ . What is the relationship between the following quantities?

$| | \overset{鈫�}{y} | |^{2}$ and $\overset{鈫�}{y} . \overset{鈫�}{x}$

Are the rows of an orthogonal matrix A necessarily orthonormal?

See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

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.