Practical linear algebra : a geometry toolbox

506 Pages·2014·5.82 MB·English

by  Gerald E.

Practical Linear Algebra
TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk
Practical Linear Algebra
A Geometry Toolbox
Third Edition
Gerald Farin
Dianne Hansford
CRC Press
Taylor & Francis Group
6000 Broken Sound Parkway NW, Suite 300
Boca Raton, FL 33487-2742
© 2014 by Taylor & Francis Group, LLC
CRC Press is an imprint of Taylor & Francis Group, an Informa business
No claim to original U.S. Government works
Version Date: 20130524
International Standard Book Number-13: 978-1-4665-7958-3 (eBook - PDF)
This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but
the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to
trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained.
If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint.
Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical,
or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without
written permission from the publishers.
For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright
Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a
variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged.
Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to
infringe.
Visit the Taylor & Francis Web site at
http://www.taylorandfrancis.com
and the CRC Press Web site at
http://www.crcpress.com
With gratitude to
Dr. James Slack and Norman Banemann.
TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk
Contents
Preface xiii
Descartes’Discovery 1 Chapter1
1.1 Local and Global Coordinates: 2D . . . . . . . . . . 2
1.2 Going from Global to Local . . . . . . . . . . . . . 6
1.3 Local and Global Coordinates: 3D . . . . . . . . . . 8
1.4 Stepping Outside the Box. . . . . . . . . . . . . . . 9
1.5 Application: Creating Coordinates. . . . . . . . . . 10
1.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 12
HereandThere: PointsandVectorsin2D 15 Chapter2
2.1 Points and Vectors . . . . . . . . . . . . . . . . . . 16
2.2 What’s the Difference? . . . . . . . . . . . . . . . . 18
2.3 Vector Fields . . . . . . . . . . . . . . . . . . . . . . 19
2.4 Length of a Vector. . . . . . . . . . . . . . . . . . . 20
2.5 Combining Points . . . . . . . . . . . . . . . . . . . 23
2.6 Independence . . . . . . . . . . . . . . . . . . . . . 26
2.7 Dot Product . . . . . . . . . . . . . . . . . . . . . . 26
2.8 Orthogonal Projections . . . . . . . . . . . . . . . . 31
2.9 Inequalities . . . . . . . . . . . . . . . . . . . . . . . 32
2.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 33
LiningUp: 2DLines 37 Chapter3
3.1 Defining a Line . . . . . . . . . . . . . . . . . . . . 38
3.2 Parametric Equation of a Line . . . . . . . . . . . . 39
3.3 Implicit Equation of a Line . . . . . . . . . . . . . . 40
3.4 Explicit Equation of a Line . . . . . . . . . . . . . . 44
vii
viii
3.5 Converting Between Parametric and Implicit
Equations . . . . . . . . . . . . . . . . . . . . . . . 44
3.5.1 Parametric to Implicit . . . . . . . . . . . . . . 45
3.5.2 Implicit to Parametric . . . . . . . . . . . . . . 45
3.6 Distance of a Point to a Line . . . . . . . . . . . . . 47
3.6.1 Starting with an Implicit Line. . . . . . . . . . 48
3.6.2 Starting with a Parametric Line . . . . . . . . 50
3.7 The Foot of a Point . . . . . . . . . . . . . . . . . . 51
3.8 A Meeting Place: Computing Intersections . . . . . 52
3.8.1 Parametric and Implicit . . . . . . . . . . . . . 53
3.8.2 Both Parametric . . . . . . . . . . . . . . . . . 55
3.8.3 Both Implicit . . . . . . . . . . . . . . . . . . . 57
3.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 58
Chapter4 ChangingShapes: LinearMapsin2D 61
4.1 Skew Target Boxes . . . . . . . . . . . . . . . . . . 62
4.2 The Matrix Form . . . . . . . . . . . . . . . . . . . 63
4.3 Linear Spaces . . . . . . . . . . . . . . . . . . . . . 66
4.4 Scalings. . . . . . . . . . . . . . . . . . . . . . . . . 68
4.5 Reflections . . . . . . . . . . . . . . . . . . . . . . . 71
4.6 Rotations . . . . . . . . . . . . . . . . . . . . . . . . 73
4.7 Shears . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.8 Projections . . . . . . . . . . . . . . . . . . . . . . . 77
4.9 Areas and Linear Maps: Determinants . . . . . . . 80
4.10 Composing Linear Maps . . . . . . . . . . . . . . . 83
4.11 More on Matrix Multiplication . . . . . . . . . . . . 87
4.12 Matrix Arithmetic Rules . . . . . . . . . . . . . . . 89
4.13 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 91
Chapter5 2×2LinearSystems 95
5.1 Skew Target Boxes Revisited . . . . . . . . . . . . . 96
5.2 The Matrix Form . . . . . . . . . . . . . . . . . . . 97
5.3 A Direct Approach: Cramer’s Rule . . . . . . . . . 98
5.4 Gauss Elimination . . . . . . . . . . . . . . . . . . . 99
5.5 Pivoting . . . . . . . . . . . . . . . . . . . . . . . . 101
5.6 Unsolvable Systems . . . . . . . . . . . . . . . . . . 104
5.7 Underdetermined Systems . . . . . . . . . . . . . . 104
5.8 Homogeneous Systems . . . . . . . . . . . . . . . . 105
5.9 Undoing Maps: Inverse Matrices . . . . . . . . . . . 107
5.10 Defining a Map . . . . . . . . . . . . . . . . . . . . 113
5.11 A Dual View . . . . . . . . . . . . . . . . . . . . . . 114
5.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 116
ix
MovingThingsAround: AffineMapsin2D 119 Chapter6
6.1 Coordinate Transformations . . . . . . . . . . . . . 120
6.2 Affine and Linear Maps . . . . . . . . . . . . . . . . 122
6.3 Translations . . . . . . . . . . . . . . . . . . . . . . 124
6.4 More General Affine Maps . . . . . . . . . . . . . . 124
6.5 Mapping Triangles to Triangles . . . . . . . . . . . 126
6.6 Composing Affine Maps . . . . . . . . . . . . . . . 128
6.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 132
EigenThings 135 Chapter7
7.1 Fixed Directions . . . . . . . . . . . . . . . . . . . . 136
7.2 Eigenvalues. . . . . . . . . . . . . . . . . . . . . . . 137
7.3 Eigenvectors . . . . . . . . . . . . . . . . . . . . . . 139
7.4 Striving for More Generality . . . . . . . . . . . . . 142
7.5 The Geometry of Symmetric Matrices . . . . . . . . 145
7.6 Quadratic Forms. . . . . . . . . . . . . . . . . . . . 149
7.7 Repeating Maps . . . . . . . . . . . . . . . . . . . . 153
7.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 155
3DGeometry 157 Chapter8
8.1 From 2D to 3D . . . . . . . . . . . . . . . . . . . . 158
8.2 Cross Product . . . . . . . . . . . . . . . . . . . . . 160
8.3 Lines . . . . . . . . . . . . . . . . . . . . . . . . . . 164
8.4 Planes . . . . . . . . . . . . . . . . . . . . . . . . . 165
8.5 Scalar Triple Product . . . . . . . . . . . . . . . . . 169
8.6 Application: Lighting and Shading . . . . . . . . . 171
8.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 174
LinearMapsin3D 177 Chapter9
9.1 Matrices and Linear Maps . . . . . . . . . . . . . . 178
9.2 Linear Spaces . . . . . . . . . . . . . . . . . . . . . 180
9.3 Scalings. . . . . . . . . . . . . . . . . . . . . . . . . 181
9.4 Reflections . . . . . . . . . . . . . . . . . . . . . . . 183
9.5 Shears . . . . . . . . . . . . . . . . . . . . . . . . . 184
9.6 Rotations . . . . . . . . . . . . . . . . . . . . . . . . 186
9.7 Projections . . . . . . . . . . . . . . . . . . . . . . . 190
9.8 Volumes and Linear Maps: Determinants . . . . . . 193
9.9 Combining Linear Maps . . . . . . . . . . . . . . . 196
9.10 Inverse Matrices . . . . . . . . . . . . . . . . . . . . 199
9.11 More on Matrices . . . . . . . . . . . . . . . . . . . 200
9.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 202