Practical Linear Algebra: A Geometry Toolbox, 4th Edition
- Length: 592 pages
- Edition: 4
- Language: English
- Publisher: A K Peters/CRC Press
- Publication Date: 2021-09-17
- ISBN-10: 0367507846
- ISBN-13: 9780367507848
- Sales Rank: #778620 (See Top 100 Books)
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.
Cover Half Title Series Page Title Page Copyright Page Dedication 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. GramSchmidt 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 Bezier 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
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