Practical Linear Algebra: A Geometry Toolbox 4th Edition by Gerald Farin, ISBN-13: 978-0367507848
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Practical Linear Algebra: A Geometry Toolbox 4th Edition by Gerald Farin, ISBN-13: 978-0367507848
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- Publisher: Chapman and Hall/CRC; 4th edition (October 13, 2021)
- Language: English
- 590 pages
- ISBN-10: 0367507846
- ISBN-13: 978-0367507848
Linear algebra is growing in importance. 3D entertainment, animations in movies and video games are developed using linear algebra. Animated characters are generated using equations straight out of this book. Linear algebra is used to extract knowledge from the massive amounts of data generated from modern technology.
The Fourth Edition of this popular text introduces linear algebra in a comprehensive, geometric, and algorithmic way. The authors start with the fundamentals in 2D and 3D, then move on to higher dimensions, expanding on the fundamentals and introducing new topics, which are necessary for many real-life applications and the development of abstract thought. Applications are introduced to motivate topics.
The subtitle, A Geometry Toolbox, hints at the book’s geometric approach, which is supported by many sketches and figures. Furthermore, the book covers applications of triangles, polygons, conics, and curves. Examples demonstrate each topic in action.
This practical approach to a linear algebra course, whether through classroom instruction or self-study, is unique to this book.
New to the Fourth Edition:
- Ten new application sections.
- A new section on change of basis. This concept now appears in several places.
- Chapters 14-16 on higher dimensions are notably revised.
- A deeper look at polynomials in the gallery of spaces.
- Introduces the QR decomposition and its relevance to least squares.
- Similarity and diagonalization are given more attention, as are eigenfunctions.
- A longer thread on least squares, running from orthogonal projections to a solution via SVD and the pseudoinverse.
- More applications for PCA have been added.
- More examples, exercises, and more on the kernel and general linear spaces.
- A list of applications has been added in Appendix A.
The book gives instructors the option of tailoring the course for the primary interests of their students: mathematics, engineering, science, computer graphics, and geometric modeling.
Table of Contents:
Cover
Half-Title Page
Series Page
Title Page
Copyright Page
Dedication Page
Contents
Preface
1 Descartes’ Discovery
1.1 Local and Global Coordinates: 2D
1.2 Going from Global to Local
1.3 Local and Global Coordinates: 3D
1.4 Stepping Outside the Box
1.5 Application: Creating Coordinates
1.6 Exercises
2 Here and There: Points and Vectors in 2D
2.1 Points and Vectors
2.2 What’s the Difference?
2.3 Vector Fields
2.4 Length of a Vector
2.5 Combining Points
2.6 Independence
2.7 Dot Product
2.8 Application: Lighting Model
2.9 Orthogonal Projections
2.10 Inequalities
2.11 Exercises
3 Lining Up: 2D Lines
3.1 Defining a Line
3.2 Parametric Equation of a Line
3.3 Implicit Equation of a Line
3.4 Explicit Equation of a Line
3.5 Converting Between Line Forms
3.6 Distance of a Point to a Line
3.7 The Foot of a Point
3.8 A Meeting Place: Computing Intersections
3.9 Application: Closest Point of Approach
3.10 Exercises
4 Changing Shapes: Linear Maps in 2D
4.1 Skew Target Boxes
4.2 The Matrix Form
4.3 Linear Spaces
4.4 Scalings
4.5 Reflections
4.6 Rotations
4.7 Shears
4.8 Projections
4.9 Application: Free-form Deformations
4.10 Areas and Linear Maps: Determinants
4.11 Composing Linear Maps
4.12 More on Matrix Multiplication
4.13 Matrix Arithmetic Rules
4.14 Exercises
5 2 × 2 Linear Systems
5.1 Skew Target Boxes Revisited
5.2 The Matrix Form
5.3 A Direct Approach: Cramer’s Rule
5.4 Gauss Elimination
5.5 Pivoting
5.6 Unsolvable Systems
5.7 Underdetermined Systems
5.8 Homogeneous Systems
5.9 Kernel
5.10 Undoing Maps: Inverse Matrices
5.11 Defining a Map
5.12 Change of Basis
5.13 Application: Intersecting Lines
5.14 Exercises
6 Moving Things Around: Affine Maps in 2D
6.1 Coordinate Transformations
6.2 Affine and Linear Maps
6.3 Translations
6.4 Application: Animation
6.5 Mapping Triangles to Triangles
6.6 Composing Affine Maps
6.7 Exercises
7 Eigen Things
7.1 Fixed Directions
7.2 Eigenvalues
7.3 Eigenvectors
7.4 Striving for More Generality
7.5 The Geometry of Symmetric Matrices
7.6 Quadratic Forms
7.7 Repeating Maps
7.8 Exercises
8 3D Geometry
8.1 From 2D to 3D
8.2 Cross Product
8.3 Lines
8.4 Planes
8.5 Scalar Triple Product
8.6 Application: Lighting and Shading
8.7 Exercises
9 Linear Maps in 3D
9.1 Matrices and Linear Maps
9.2 Linear Spaces
9.3 Scalings
9.4 Reflections
9.5 Shears
9.6 Rotations
9.7 Projections
9.8 Volumes and Linear Maps: Determinants
9.9 Combining Linear Maps
9.10 Inverse Matrices
9.11 Application: Mapping Normals
9.12 More on Matrices
9.13 Exercises
10 Affine Maps in 3D
10.1 Affine Maps
10.2 Translations
10.3 Mapping Tetrahedra
10.4 Parallel Projections
10.5 Homogeneous Coordinates and Perspective Maps
10.6 Application: Building Instance Models
10.7 Exercises
11 Interactions in 3D
11.1 Distance Between a Point and a Plane
11.2 Distance Between Two Lines
11.3 Lines and Planes: Intersections
11.4 Intersecting a Triangle and a Line
11.5 Reflections
11.6 Intersecting Three Planes
11.7 Intersecting Two Planes
11.8 Creating Orthonormal Coordinate Systems
11.9 Application: Camera Model
11.10 Exercises
12 Gauss for Linear Systems
12.1 The Problem
12.2 The Solution via Gauss Elimination
12.3 Homogeneous Linear Systems
12.4 Inverse Matrices
12.5 LU Decomposition
12.6 Determinants
12.7 Least Squares
12.8 Application: Fitting Data from a Femoral Head
12.9 Exercises
13 Alternative System Solvers
13.1 The Householder Method
13.2 Vector Norms
13.3 Matrix Norms
13.4 The Condition Number
13.5 Vector Sequences
13.6 Iterative Methods: Gauss-Jacobi and Gauss-Seidel
13.7 Application: Mesh Smoothing
13.8 Exercises
14 General Linear Spaces
14.1 Basic Properties of Linear Spaces
14.2 Linear Maps
14.3 Inner Products
14.4 Gram-Schmidt Orthonormalization
14.5 QR Decompositon
14.6 A Gallery of Spaces
14.7 Least Squares
14.8 Application: Music Analysis
14.9 Exercises
15 Eigen Things Revisited
15.1 The Basics Revisited
15.2 Similarity and Diagonalization
15.3 Quadratic Forms
15.4 The Power Method
15.5 Application: Google Eigenvector
15.6 QR Algorithm
15.7 Eigenfunctions
15.8 Application: Influenza Modeling
15.9 Exercises
16 The Singular Value Decomposition
16.1 The Geometry of the 2 × 2 Cases
16.2 The General Case
16.3 SVD Steps
16.4 Singular Values and Volumes
16.5 The Pseudoinverse
16.6 Least Squares
16.7 Application: Image Compression
16.8 Principal Component Analysis
16.9 Application: Face Authentication
16.10 Exercises
17 Breaking It Up: Triangles
17.1 Barycentric Coordinates
17.2 Affine Invariance
17.3 Some Special Points
17.4 2D Triangulations
17.5 A Data Structure
17.6 Application: Point Location
17.7 3D Triangulations
17.8 Exercises
18 Putting Lines Together: Polylines and Polygons
18.1 Polylines
18.2 Polygons
18.3 Convexity
18.4 Types of Polygons
18.5 Unusual Polygons
18.6 Turning Angles and Winding Numbers
18.7 Area
18.8 Application: Planarity Test
18.9 Application: Inside or Outside?
18.10 Exercises
19 Conics
19.1 The General Conic
19.2 Analyzing Conics
19.3 General Conic to Standard Position
19.4 The Action Ellipse
19.5 Exercises
20 Curves
20.1 Parametric Curves
20.2 Properties of Bézier Curves
20.3 The Matrix Form
20.4 Derivatives
20.5 Composite Curves
20.6 The Geometry of Planar Curves
20.7 Application: Moving along a Curve
20.8 Exercises
A Applications
B Glossary
C Selected Exercise Solutions
Bibliography
Index
Gerald Farin (deceased) was a professor in the School of Computing, Informatics, and Design Systems Engineering (CIDSE) at Arizona State University. He received his doctoral degree in mathematics from the University of Braunschweig, Germany. His extensive experience in geometric design started at Daimler-Benz. He was a founding member of the editorial board for the journal Computer-Aided Geometric Design (Elsevier), and he served as co-editor in chief for more than 20 years. He published more than 100 research papers. Gerald also organized numerous conferences and authored or edited 29 books. This includes his much read and referenced textbook Curves and Surfaces for CAGD and his book on NURBS. In addition to this book, Gerald and Dianne co-authored The Essentials of CAGD, Mathematical Principles for Scientific Computing and Visualization both also published by AK Peters/CRC Press.
Dianne Hansford, received her Ph.D. from Arizona State University. Her research interests are in the field of geometric modeling with a focus on industrial curve and surface applications related to mathematical definitions of shape. Together with Gerald Farin (deceased), she delivered custom software solutions, advisement on best practices, and taught on-site courses as a consultant. She is a co-founder of 3D Compression Technologies. She is now lecturer in the School of Computing, Informatics, and Design Systems Engineering (CIDSE) at Arizona State University, primarily teaching geometric design, computer graphics, and scientific computing and visualization. In addition to this book, Gerald and Dianne co-authored The Essentials of CAGD, Mathematical Principles for Scientific Computing and Visualization both also published by AK Peters/CRC Press.
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