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Searching... | 30000010076905 | QA184.2 F37 2005 | Open Access Book | Book | Searching... |
Searching... | 30000010076904 | QA184.2 F37 2005 | Open Access Book | Book | Searching... |
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Summary
Summary
Practical Linear Algebra introduces students in math, science, engineering, and computer science to Linear Algebra from an intuitive and geometric viewpoint, creating a level of understanding that goes far beyond mere matrix manipulations. Practical aspects, such as computer graphics topics and numerical strategies, are covered throughout, and thus students can build a "Geometry Toolbox," based on a geometric understanding of the key concepts. This book covers all the standard linear algebra material for a first-year course; the authors teach by motivation, illustration, and example rather than by using a theorem/proof style. Special Features: - Clear visual representations (more than 200 figures) for improved material comprehension. - Hand-drawn sketches encourage students to create their own sketches when solving problems-developing another layer of learning. - Numerous examples show applications to real-world problems. - Problems at the end of each chapter allow students to test their understanding of the material presented. Solutions to selected problems are provided. - Concise chapter summaries highlight the most important points, giving students focus for their approach to learning. An instructor's manual will be available soon.
Reviews 1
Choice Review
Is linear algebra a tool that is capable of solving a myriad of problems or an example of mathematical structures and reasoning? The answer, of course, is both. In this book, Farin and Hansford certainly take the first viewpoint. Common theorem-proof presentation has been replaced by motivation, examples, or graphics. Their goal is to give the student an intuitive and geometric grasp of the fundamental concepts. The book is aimed at the freshman/sophomore level and is quite appropriate for students in engineering and computer graphics as well as in mathematics. It is well written and the examples are carefully chosen to motivate or exemplify the topic at hand. As a consequence, it is significantly different from a traditional work in the area. For example, "eigen things" (their term) are introduced in the plane and used to diagonalize symmetric matrices, but the general problem of diagonalizing a matrix is omitted. There is no mention of similar matrices. It appears to be a very good book, if the goal is introduction to applications, but perhaps not for introduction to mathematical proofs. ^BSumming Up: Recommended. Lower-division undergraduates; two-year technical program students. J. R. Burke Gonzaga University
Table of Contents
Preface | p. xiii |
Chapter 1 Descartes' Discovery | p. 1 |
1.1 Local and Global Coordinates: 2D | p. 2 |
1.2 Going from Global to Local | p. 6 |
1.3 Local and Global Coordinates: 3D | p. 8 |
1.4 Stepping Outside the Box | p. 9 |
1.5 Creating Coordinates | p. 10 |
1.6 Exercises | p. 12 |
Chapter 2 Here and There: Points and Vectors in 2D | p. 13 |
2.1 Points and Vectors | p. 14 |
2.2 What's the Difference? | p. 16 |
2.3 Vector Fields | p. 17 |
2.4 Length of a Vector | p. 18 |
2.5 Combining Points | p. 21 |
2.6 Independence | p. 24 |
2.7 Dot Product | p. 25 |
2.8 Orthogonal Projections | p. 29 |
2.9 Inequalities | p. 30 |
2.10 Exercises | p. 31 |
Chapter 3 Lining Up: 2D Lines | p. 33 |
3.1 Defining a Line | p. 34 |
3.2 Parametric Equation of a Line | p. 35 |
3.3 Implicit Equation of a Line | p. 37 |
3.4 Explicit Equation of a Line | p. 40 |
3.5 Converting Between Parametric and Implicit Equations | p. 41 |
3.6 Distance of a Point to a Line | p. 43 |
3.7 The Foot of a Point | p. 47 |
3.8 A Meeting Place: Computing Intersections | p. 48 |
3.9 Exercises | p. 54 |
Chapter 4 Changing Shapes: Linear Maps in 2D | p. 57 |
4.1 Skew Target Boxes | p. 58 |
4.2 The Matrix Form | p. 59 |
4.3 More about Matrices | p. 61 |
4.4 Scalings | p. 63 |
4.5 Reflections | p. 65 |
4.6 Rotations | p. 68 |
4.7 Shears | p. 69 |
4.8 Projections | p. 71 |
4.9 The Kernel of a Projection | p. 73 |
4.10 Areas and Linear Maps: Determinants | p. 74 |
4.11 Composing Linear Maps | p. 77 |
4.12 More on Matrix Multiplication | p. 81 |
4.13 Working with Matrices | p. 83 |
4.14 Exercises | p. 84 |
Chapter 5 2 x 2 Linear Systems | p. 87 |
5.1 Skew Target Boxes Revisited | p. 88 |
5.2 The Matrix Form | p. 89 |
5.3 A Direct Approach: Cramer's Rule | p. 90 |
5.4 Gauss Elimination | p. 91 |
5.5 Undoing Maps: Inverse Matrices | p. 93 |
5.6 Unsolvable Systems | p. 99 |
5.7 Underdetermined Systems | p. 100 |
5.8 Homogeneous Systems | p. 100 |
5.9 Numerical Strategies: Pivoting | p. 102 |
5.10 Defining a Map | p. 104 |
5.11 Exercises | p. 104 |
Chapter 6 Moving Things Around: Affine Maps in 2D | p. 107 |
6.1 Coordinate Transformations | p. 108 |
6.2 Affine and Linear Maps | p. 110 |
6.3 Translations | p. 111 |
6.4 More General Affine Maps | p. 112 |
6.5 Mapping Triangles to Triangles | p. 114 |
6.6 Composing Affine Maps | p. 116 |
6.7 Exercises | p. 120 |
Chapter 7 Eigen Things | p. 123 |
7.1 Fixed Directions | p. 124 |
7.2 Eigenvalues | p. 125 |
7.3 Eigenvectors | p. 127 |
7.4 Special Cases | p. 129 |
7.5 The Geometry of Symmetric Matrices | p. 132 |
7.6 Repeating Maps | p. 135 |
7.7 The Condition of a Map | p. 137 |
7.8 Exercises | p. 138 |
Chapter 8 Breaking It Up: Triangles | p. 141 |
8.1 Barycentric Coordinates | p. 142 |
8.2 Affine Invariance | p. 144 |
8.3 Some Special Points | p. 145 |
8.4 2D Triangulations | p. 148 |
8.5 A Data Structure | p. 149 |
8.6 Point Location | p. 150 |
8.7 3D Triangulations | p. 151 |
8.8 Exercises | p. 153 |
Chapter 9 Conics | p. 155 |
9.1 The General Conic | p. 156 |
9.2 Analyzing Conics | p. 160 |
9.3 The Position of a Conic | p. 162 |
9.4 Exercises | p. 163 |
Chapter 10 3D Geometry | p. 165 |
10.1 From 2D to 3D | p. 166 |
10.2 Cross Product | p. 168 |
10.3 Lines | p. 172 |
10.4 Planes | p. 173 |
10.5 Application: Lighting and Shading | p. 177 |
10.6 Scalar Triple Product | p. 180 |
10.7 Linear Spaces | p. 181 |
10.8 Exercises | p. 183 |
Chapter 11 Interactions in 3D | p. 185 |
11.1 Distance Between a Point and a Plane | p. 186 |
11.2 Distance Between Two Lines | p. 187 |
11.3 Lines and Planes: Intersections | p. 189 |
11.4 Intersecting a Triangle and a Line | p. 191 |
11.5 Lines and Planes: Reflections | p. 191 |
11.6 Intersecting Three Planes | p. 193 |
11.7 Intersecting Two Planes | p. 194 |
11.8 Creating Orthonormal Coordinate Systems | p. 195 |
11.9 Exercises | p. 197 |
Chapter 12 Linear Maps in 3D | p. 199 |
12.1 Matrices and Linear Maps | p. 200 |
12.2 Scalings | p. 202 |
12.3 Reflections | p. 204 |
12.4 Shears | p. 204 |
12.5 Projections | p. 207 |
12.6 Rotations | p. 209 |
12.7 Volumes and Linear Maps: Determinants | p. 213 |
12.8 Combining Linear Maps | p. 216 |
12.9 More on Matrices | p. 218 |
12.10 Inverse Matrices | p. 219 |
12.11 Exercises | p. 221 |
Chapter 13 Affine Maps in 3D | p. 223 |
13.1 Affine Maps | p. 224 |
13.2 Translations | p. 225 |
13.3 Mapping Tetrahedra | p. 225 |
13.4 Projections | p. 229 |
13.5 Homogeneous Coordinates and Perspective Maps | p. 232 |
13.6 Exercises | p. 238 |
Chapter 14 General Linear Systems | p. 241 |
14.1 The Problem | p. 242 |
14.2 The Solution via Gauss Elimination | p. 244 |
14.3 Determinants | p. 250 |
14.4 Overdetermined Systems | p. 253 |
14.5 Inverse Matrices | p. 256 |
14.6 LU Decomposition | p. 258 |
14.7 Exercises | p. 262 |
Chapter 15 General Linear Spaces | p. 265 |
15.1 Basic Properties | p. 266 |
15.2 Linear Maps | p. 268 |
15.3 Inner Products | p. 271 |
15.4 Gram-Schmidt Orthonormalization | p. 271 |
15.5 Higher Dimensional Eigen Things | p. 272 |
15.6 A Gallery of Spaces | p. 274 |
15.7 Exercises | p. 276 |
Chapter 16 Numerical Methods | p. 279 |
16.1 Another Linear System Solver: The Householder Method | p. 280 |
16.2 Vector Norms and Sequences | p. 285 |
16.3 Iterative System Solvers: Gauss-Jacobi and Gauss-Seidel | p. 287 |
16.4 Finding Eigenvalues: the Power Method | p. 290 |
16.5 Exercises | p. 294 |
Chapter 17 Putting Lines Together: Polylines and Polygons | p. 297 |
17.1 Polylines | p. 298 |
17.2 Polygons | p. 299 |
17.3 Convexity | p. 300 |
17.4 Types of Polygons | p. 301 |
17.5 Unusual Polygons | p. 302 |
17.6 Turning Angles and Winding Numbers | p. 304 |
17.7 Area | p. 305 |
17.8 Planarity Test | p. 309 |
17.9 Inside or Outside? | p. 310 |
17.10 Exercises | p. 313 |
Chapter 18 Curves | p. 315 |
18.1 Application: Parametric Curves | p. 316 |
18.2 Properties of Bezier Curves | p. 321 |
18.3 The Matrix Form | p. 323 |
18.4 Derivatives | p. 324 |
18.5 Composite Curves | p. 326 |
18.6 The Geometry of Planar Curves | p. 327 |
18.7 Moving along a Curve | p. 329 |
18.8 Exercises | p. 331 |
Appendix A PostScript Tutorial | p. 333 |
A.1 A Warm-Up Example | p. 333 |
A.2 Overview | p. 336 |
A.3 Affine Maps | p. 338 |
A.4 Variables | p. 339 |
A.5 Loops | p. 340 |
A.6 CTM | p. 341 |
Appendix B Selected Problem Solutions | p. 343 |
Glossary | p. 367 |
Bibliography | p. 371 |
Index | p. 373 |