Practical Linear Algebra: A Geometry Toolbox (Textbooks in Mathematics)

"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"-- Provided by publisher 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. Cover 1 Half Title 2 Series Page 3 Title Page 4 Copyright Page 5 Dedication 6 Contents 8 Preface 14 1. Descartes’ Discovery 20 1.1. Local and Global Coordinates: 2D 21 1.2. Going from Global to Local 25 1.3. Local and Global Coordinates: 3D 27 1.4. Stepping Outside the Box 28 1.5. Application: Creating Coordinates 29 1.6. Exercises 31 2. Here and There: Points and Vectors in 2D 34 2.1. Points and Vectors 35 2.2. What’s the Difference? 37 2.3. Vector Fields 38 2.4. Length of a Vector 39 2.5. Combining Points 42 2.6. Independence 45 2.7. Dot Product 47 2.8. Application: Lighting Model 51 2.9. Orthogonal Projections 52 2.10. Inequalities 54 2.11. Exercises 55 3. Lining Up: 2D Lines 60 3.1. Defining a Line 61 3.2. Parametric Equation of a Line 62 3.3. Implicit Equation of a Line 63 3.4. Explicit Equation of a Line 67 3.5. Converting Between Line Forms 67 3.6. Distance of a Point to a Line 70 3.7. The Foot of a Point 74 3.8. A Meeting Place: Computing Intersections 75 3.9. Application: Closest Point of Approach 81 3.10. Exercises 82 4. Changing Shapes: Linear Maps in 2D 86 4.1. Skew Target Boxes 87 4.2. The Matrix Form 88 4.3. Linear Spaces 92 4.4. Scalings 94 4.5. Reflections 96 4.6. Rotations 98 4.7. Shears 100 4.8. Projections 102 4.9. Application: Free-form Deformations 105 4.10. Areas and Linear Maps: Determinants 107 4.11. Composing Linear Maps 109 4.12. More on Matrix Multiplication 113 4.13. Matrix Arithmetic Rules 115 4.14. Exercises 117 5. 2 × 2 Linear Systems 120 5.1. Skew Target Boxes Revisited 121 5.2. The Matrix Form 122 5.3. A Direct Approach: Cramer’s Rule 123 5.4. Gauss Elimination 124 5.5. Pivoting 126 5.6. Unsolvable Systems 129 5.7. Underdetermined Systems 129 5.8. Homogeneous Systems 130 5.9. Kernel 132 5.10. Undoing Maps: Inverse Matrices 134 5.11. Defining a Map 140 5.12. Change of Basis 142 5.13. Application: Intersecting Lines 148 5.14. Exercises 150 6. Moving Things Around: Affine Maps in 2D 154 6.1. Coordinate Transformations 155 6.2. Affine and Linear Maps 157 6.3. Translations 159 6.4. Application: Animation 159 6.5. Mapping Triangles to Triangles 162 6.6. Composing Affine Maps 164 6.7. Exercises 169 7. Eigen Things 172 7.1. Fixed Directions 173 7.2. Eigenvalues 174 7.3. Eigenvectors 176 7.4. Striving for More Generality 179 7.5. The Geometry of Symmetric Matrices 182 7.6. Quadratic Forms 185 7.7. Repeating Maps 191 7.8. Exercises 193 8. 3D Geometry 196 8.1. From 2D to 3D 197 8.2. Cross Product 199 8.3. Lines 204 8.4. Planes 205 8.5. Scalar Triple Product 209 8.6. Application: Lighting and Shading 210 8.7. Exercises 214 9. Linear Maps in 3D 216 9.1. Matrices and Linear Maps 217 9.2. Linear Spaces 219 9.3. Scalings 220 9.4. Reflections 222 9.5. Shears 223 9.6. Rotations 225 9.7. Projections 229 9.8. Volumes and Linear Maps: Determinants 232 9.9. Combining Linear Maps 236 9.10. Inverse Matrices 238 9.11. Application: Mapping Normals 239 9.12. More on Matrices 241 9.13. Exercises 243 10. Affine Maps in 3D 248 10.1. Affine Maps 249 10.2. Translations 250 10.3. Mapping Tetrahedra 250 10.4. Parallel Projections 254 10.5. Homogeneous Coordinates and Perspective Maps 258 10.6. Application: Building Instance Models 262 10.7. Exercises 265 11. Interactions in 3D 268 11.1. Distance Between a Point and a Plane 269 11.2. Distance Between Two Lines 270 11.3. Lines and Planes: Intersections 272 11.4. Intersecting a Triangle and a Line 274 11.5. Reflections 275 11.6. Intersecting Three Planes 276 11.7. Intersecting Two Planes 277 11.8. Creating Orthonormal Coordinate Systems 278 11.9. Application: Camera Model 281 11.10. Exercises 283 12. Gauss for Linear Systems 286 12.1. The Problem 287 12.2. The Solution via Gauss Elimination 290 12.3. Homogeneous Linear Systems 298 12.4. Inverse Matrices 300 12.5. LU Decomposition 303 12.6. Determinants 307 12.7. Least Squares 310 12.8. Application: Fitting Data from a Femoral Head 315 12.9. Exercises 317 13. Alternative System Solvers 322 13.1. The Householder Method 323 13.2. Vector Norms 330 13.3. Matrix Norms 333 13.4. The Condition Number 336 13.5. Vector Sequences 339 13.6. Iterative Methods: Gauss-Jacobi and Gauss-Seidel 341 13.7. Application: Mesh Smoothing 345 13.8. Exercises 347 14. General Linear Spaces 350 14.1. Basic Properties of Linear Spaces 351 14.2. Linear Maps 355 14.3. Inner Products 358 14.4. GramSchmidt 363 14.5. QR Decompositon 365 14.6. A Gallery of Spaces 368 14.7. Least Squares 373 14.8. Application: Music Analysis 376 14.9. Exercises 379 15. Eigen Things Revisited 384 15.1. The Basics Revisited 385 15.2. Similarity and Diagonalization 393 15.3. Quadratic Forms 397 15.4. The Power Method 399 15.5. Application: Google Eigenvector 403 15.6. QR Algorithm 406 15.7. Eigenfunctions 407 15.8. Application: Influenza Modeling 409 15.9. Exercises 411 16. The Singular Value Decomposition 416 16.1. The Geometry of the 2 × 2 Cases 417 16.2. The General Case 421 16.3. SVD Steps 426 16.4. Singular Values and Volumes 427 16.5. The Pseudoinverse 427 16.6. Least Squares 429 16.7. Application: Image Compression 433 16.8. Principal Component Analysis 436 16.9. Application: Face Authentication 442 16.10. Exercises 444 17. Breaking It Up: Triangles 448 17.1. Barycentric Coordinates 449 17.2. Affine Invariance 452 17.3. Some Special Points 452 17.4. 2D Triangulations 455 17.5. A Data Structure 456 17.6. Application: Point Location 458 17.7. 3D Triangulations 459 17.8. Exercises 460 18. Putting Lines Together: Polylines and Polygons 462 18.1. Polylines 463 18.2. Polygons 464 18.3. Convexity 465 18.4. Types of Polygons 466 18.5. Unusual Polygons 467 18.6. Turning Angles and Winding Numbers 469 18.7. Area 470 18.8. Application: Planarity Test 474 18.9. Application: Inside or Outside? 475 18.10. Exercises 478 19. Conics 480 19.1. The General Conic 481 19.2. Analyzing Conics 486 19.3. General Conic to Standard Position 487 19.4. The Action Ellipse 490 19.5. Exercises 491 20. Curves 494 20.1. Parametric Curves 495 20.2. Properties of Bezier Curves 500 20.3. The Matrix Form 501 20.4. Derivatives 503 20.5. Composite Curves 505 20.6. The Geometry of Planar Curves 505 20.7. Application: Moving along a Curve 507 20.8. Exercises 510 A. Applications 512 B. Glossary 514 C. Selected Exercise Solutions 530 Bibliography 578 Index 580 A;,Algebra;,Geometry;,Linear;,Practical;,Toolbox A,Algebra,Geometry,Linear,Practical,Toolbox "Practical Linear Algebra covers all the concepts in a traditional undergraduate-level linear algebra course, but with a focus on practical applications. The book develops these fundamental concepts in 2D and 3D with a strong emphasis on geometric understanding before presenting the general (n-dimensional) concept. The book does not employ a theorem/proof structure, and it spends very little time on tedious, by-hand calculations (e.g., reduction to row-echelon form), which in most job applications are performed by products such as Mathematica. Instead the book presents concepts through examples and applications. "--