**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 BÃ©zier 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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