Practical Linear Algebra: A Geometry Toolbox

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.

1. Descartes' Discovery
2. Here and There: Points and Vectors in 2D
3. Lining Up: 2D Lines
4. Changing Shapes: Linear Maps in 2D
5. 2 2 Linear Systems
6. Moving Things Around: Affine Maps in 2D
7. Eigen Things
8. 3D Geometry
9. Linear Maps in 3D
10. Affine Maps in 3D
11. Interactions in 3D
12. Gauss or Linear Systems
13. Alternative System Solvers
14. General Linear Spaces
15. Eigen Things Revisited
16. The Singular Value Decomposition
17. Breaking It Up: Triangles
18. Putting Lines Together: Polylines and Polygons
19. Conics
20. Curves

Appendices

1. Applications
2. Glossary
3. Select Exercise Solutions

Bibliography

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.