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The Manga Guide to Linear Algebra by Ltd. Trend-Pro Co., Iroha Inoue, Shin Takahashi

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The Manga Guide
to
LINEAR ALGEBRA
Shin Takahashi,
Iroha Inoue, and
Trend-Pro Co., Ltd.
The Manga Guide to Linear Algebra.
Copyright © 2012 by Shin Takahashi and TREND-PRO Co., Ltd.
The Manga Guide to Linear Algebra is a translation of the Japanese original, Manga de wakaru
senkeidaisuu, published by Ohmsha, Ltd. of Tokyo, Japan, © 2008 by Shin Takahashi and TREND-
PRO Co., Ltd.
This English edition is co-published by No Starch Press, Inc. and Ohmsha, Ltd.
All rights reserved. No part of this work may be reproduced or transmitted in any form or by any
means, electronic or mechanical, including photocopying, recording, or by any information storage
or retrieval system, without the prior written permission of the copyright owner and the publisher.
First printing
16 15 14 13 12 1 2 3 4 5 6 7 8 9
ISBN-10: 1-59327-413-0
ISBN-13: 978-1-59327-413-9
Publisher: William Pollock
Author: Shin Takahashi
Illustrator: Iroha Inoue
Producer: TREND-PRO Co., Ltd.
Production Editor: Alison Law
Developmental Editor: Keith Fancher
Translator: Fredrik Lindh
Technical Reviewer: Eric Gossett
Compositor: Riley Hoffman
Proofreader: Paula L. Fleming
Indexer: BIM Indexing & Proofreading Services
For information on book distributors or translations, please contact No Starch Press, Inc. directly:
No Starch Press, Inc.
38 Ringold Street, San Francisco, CA 94103
phone: 415.863.9900; fax: 415.863.9950; info@nostarch.com; http://www.nostarch.com/
Library of Congress Cataloging-in-Publication Data
Takahashi, Shin.
[Manga de wakaru senkei daisu. English]
The manga guide to linear algebra / Shin Takahashi, Iroha Inoue, Trend-pro Co. Ltd.
p. cm.
ISBN 978-1-59327-413-9 (pbk.) -- ISBN 1-59327-413-0 (pbk.)
1. Algebras, Linear--Comic books, strips, etc. 2. Graphic novels. I. Inoue, Iroha. II. Trend-pro Co.
III. Title.
QA184.2.T3513 2012
512’.50222--dc23
2012012824
No Starch Press and the No Starch Press logo are registered trademarks of No Starch Press, Inc.
Other product and company names mentioned herein may be the trademarks of their respective own-
ers. Rather than use a trademark symbol with every occurrence of a trademarked name, we are using
the names only in an editorial fashion and to the benefit of the trademark owner, with no intention of
infringement of the trademark.
The information in this book is distributed on an “As Is” basis, without warranty. While every pre-
caution has been taken in the preparation of this work, neither the author nor No Starch Press, Inc.
shall have any liability to any person or entity with respect to any loss or damage caused or alleged
to be caused directly or indirectly by the information contained in it.
All characters in this publication are fictitious, and any resemblance to real persons, living or dead,
is purely coincidental.
Contents
Preface ....................................................... xi
Prologue
Let the Training Begin! ..........................................1
1
What Is Linear Algebra? ........................................9
An Overview of Linear Algebra.......................................14
2
The Fundamentals ..............................................21
Number Systems .................................................25
Implication and Equivalence ........................................27
Propositions .................................................27
Implication ..................................................28
Equivalence ..................................................29
Set Theory ......................................................30
Sets ........................................................30
Set Symbols..................................................32
Subsets .....................................................33
Functions .......................................................35
Images ......................................................40
Domain and Range ............................................44
Onto and One-to-One Functions ..................................46
Inverse Functions .............................................48
Linear Transformations.........................................50
Combinations and Permutations.....................................55
Not All “Rules for Ordering” Are Functions ............................61
3
Intro to Matrices .............................................63
What Is a Matrix? .................................................66
Matrix Calculations ...............................................70
Addition .....................................................70
Subtraction ..................................................71
Scalar Multiplication ...........................................72
Matrix Multiplication...........................................73
Special Matrices..................................................77
Zero Matrices.................................................77
viii Contents
Transpose Matrices ............................................78
Symmetric Matrices ...........................................79
Upper Triangular and Lower Triangular Matrices .....................79
Diagonal Matrices .............................................80
Identity Matrices ..............................................82
4
More Matrices.................................................85
Inverse Matrices..................................................86
Calculating Inverse Matrices ........................................88
Determinants ....................................................95
Calculating Determinants ..........................................96
Calculating Inverse Matrices Using Cofactors .........................108
M
ij
. .......................................................108
C
ij
. .......................................................109
Calculating Inverse Matrices....................................110
Using Determinants ..............................................111
Solving Linear Systems with Cramer’s Rule ...........................111
5
Introduction to Vectors .....................................113
What Are Vectors? ...............................................116
Vector Calculations ..............................................125
Geometric Interpretations .........................................127
6
More Vectors ................................................131
Linear Independence .............................................132
Bases .........................................................140
Dimension .....................................................149
Subspaces ..................................................150
Basis and Dimension .........................................156
Coordinates ....................................................161
7
Linear Transformations.......................................163
What Is a Linear Transformation?...................................166
Why We Study Linear Transformations...............................173
Special Transformations ..........................................178
Scaling .....................................................179
Rotation ....................................................180
Translation .................................................182
3-D Projection ...............................................185

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