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Cover image for Practical linear algebra : a geometry toolbox
Title:
Practical linear algebra : a geometry toolbox
Personal Author:
Publication Information:
Wellesley, Mass. : A K Peters, 2005.
Physical Description:
xvi, 384 p. : ill. ; 25 cm.
ISBN:
9781568812342
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30000010076905 QA184.2 F37 2005 Open Access Book Book
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30000010076904 QA184.2 F37 2005 Open Access Book Book
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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 Common theorem-proof presentation has been replaced by motivation, examples, or graphics.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

Prefacep. xiii
Chapter 1 Descartes' Discoveryp. 1
1.1 Local and Global Coordinates: 2Dp. 2
1.2 Going from Global to Localp. 6
1.3 Local and Global Coordinates: 3Dp. 8
1.4 Stepping Outside the Boxp. 9
1.5 Creating Coordinatesp. 10
1.6 Exercisesp. 12
Chapter 2 Here and There: Points and Vectors in 2Dp. 13
2.1 Points and Vectorsp. 14
2.2 What's the Difference?p. 16
2.3 Vector Fieldsp. 17
2.4 Length of a Vectorp. 18
2.5 Combining Pointsp. 21
2.6 Independencep. 24
2.7 Dot Productp. 25
2.8 Orthogonal Projectionsp. 29
2.9 Inequalitiesp. 30
2.10 Exercisesp. 31
Chapter 3 Lining Up: 2D Linesp. 33
3.1 Defining a Linep. 34
3.2 Parametric Equation of a Linep. 35
3.3 Implicit Equation of a Linep. 37
3.4 Explicit Equation of a Linep. 40
3.5 Converting Between Parametric and Implicit Equationsp. 41
3.6 Distance of a Point to a Linep. 43
3.7 The Foot of a Pointp. 47
3.8 A Meeting Place: Computing Intersectionsp. 48
3.9 Exercisesp. 54
Chapter 4 Changing Shapes: Linear Maps in 2Dp. 57
4.1 Skew Target Boxesp. 58
4.2 The Matrix Formp. 59
4.3 More about Matricesp. 61
4.4 Scalingsp. 63
4.5 Reflectionsp. 65
4.6 Rotationsp. 68
4.7 Shearsp. 69
4.8 Projectionsp. 71
4.9 The Kernel of a Projectionp. 73
4.10 Areas and Linear Maps: Determinantsp. 74
4.11 Composing Linear Mapsp. 77
4.12 More on Matrix Multiplicationp. 81
4.13 Working with Matricesp. 83
4.14 Exercisesp. 84
Chapter 5 2 x 2 Linear Systemsp. 87
5.1 Skew Target Boxes Revisitedp. 88
5.2 The Matrix Formp. 89
5.3 A Direct Approach: Cramer's Rulep. 90
5.4 Gauss Eliminationp. 91
5.5 Undoing Maps: Inverse Matricesp. 93
5.6 Unsolvable Systemsp. 99
5.7 Underdetermined Systemsp. 100
5.8 Homogeneous Systemsp. 100
5.9 Numerical Strategies: Pivotingp. 102
5.10 Defining a Mapp. 104
5.11 Exercisesp. 104
Chapter 6 Moving Things Around: Affine Maps in 2Dp. 107
6.1 Coordinate Transformationsp. 108
6.2 Affine and Linear Mapsp. 110
6.3 Translationsp. 111
6.4 More General Affine Mapsp. 112
6.5 Mapping Triangles to Trianglesp. 114
6.6 Composing Affine Mapsp. 116
6.7 Exercisesp. 120
Chapter 7 Eigen Thingsp. 123
7.1 Fixed Directionsp. 124
7.2 Eigenvaluesp. 125
7.3 Eigenvectorsp. 127
7.4 Special Casesp. 129
7.5 The Geometry of Symmetric Matricesp. 132
7.6 Repeating Mapsp. 135
7.7 The Condition of a Mapp. 137
7.8 Exercisesp. 138
Chapter 8 Breaking It Up: Trianglesp. 141
8.1 Barycentric Coordinatesp. 142
8.2 Affine Invariancep. 144
8.3 Some Special Pointsp. 145
8.4 2D Triangulationsp. 148
8.5 A Data Structurep. 149
8.6 Point Locationp. 150
8.7 3D Triangulationsp. 151
8.8 Exercisesp. 153
Chapter 9 Conicsp. 155
9.1 The General Conicp. 156
9.2 Analyzing Conicsp. 160
9.3 The Position of a Conicp. 162
9.4 Exercisesp. 163
Chapter 10 3D Geometryp. 165
10.1 From 2D to 3Dp. 166
10.2 Cross Productp. 168
10.3 Linesp. 172
10.4 Planesp. 173
10.5 Application: Lighting and Shadingp. 177
10.6 Scalar Triple Productp. 180
10.7 Linear Spacesp. 181
10.8 Exercisesp. 183
Chapter 11 Interactions in 3Dp. 185
11.1 Distance Between a Point and a Planep. 186
11.2 Distance Between Two Linesp. 187
11.3 Lines and Planes: Intersectionsp. 189
11.4 Intersecting a Triangle and a Linep. 191
11.5 Lines and Planes: Reflectionsp. 191
11.6 Intersecting Three Planesp. 193
11.7 Intersecting Two Planesp. 194
11.8 Creating Orthonormal Coordinate Systemsp. 195
11.9 Exercisesp. 197
Chapter 12 Linear Maps in 3Dp. 199
12.1 Matrices and Linear Mapsp. 200
12.2 Scalingsp. 202
12.3 Reflectionsp. 204
12.4 Shearsp. 204
12.5 Projectionsp. 207
12.6 Rotationsp. 209
12.7 Volumes and Linear Maps: Determinantsp. 213
12.8 Combining Linear Mapsp. 216
12.9 More on Matricesp. 218
12.10 Inverse Matricesp. 219
12.11 Exercisesp. 221
Chapter 13 Affine Maps in 3Dp. 223
13.1 Affine Mapsp. 224
13.2 Translationsp. 225
13.3 Mapping Tetrahedrap. 225
13.4 Projectionsp. 229
13.5 Homogeneous Coordinates and Perspective Mapsp. 232
13.6 Exercisesp. 238
Chapter 14 General Linear Systemsp. 241
14.1 The Problemp. 242
14.2 The Solution via Gauss Eliminationp. 244
14.3 Determinantsp. 250
14.4 Overdetermined Systemsp. 253
14.5 Inverse Matricesp. 256
14.6 LU Decompositionp. 258
14.7 Exercisesp. 262
Chapter 15 General Linear Spacesp. 265
15.1 Basic Propertiesp. 266
15.2 Linear Mapsp. 268
15.3 Inner Productsp. 271
15.4 Gram-Schmidt Orthonormalizationp. 271
15.5 Higher Dimensional Eigen Thingsp. 272
15.6 A Gallery of Spacesp. 274
15.7 Exercisesp. 276
Chapter 16 Numerical Methodsp. 279
16.1 Another Linear System Solver: The Householder Methodp. 280
16.2 Vector Norms and Sequencesp. 285
16.3 Iterative System Solvers: Gauss-Jacobi and Gauss-Seidelp. 287
16.4 Finding Eigenvalues: the Power Methodp. 290
16.5 Exercisesp. 294
Chapter 17 Putting Lines Together: Polylines and Polygonsp. 297
17.1 Polylinesp. 298
17.2 Polygonsp. 299
17.3 Convexityp. 300
17.4 Types of Polygonsp. 301
17.5 Unusual Polygonsp. 302
17.6 Turning Angles and Winding Numbersp. 304
17.7 Areap. 305
17.8 Planarity Testp. 309
17.9 Inside or Outside?p. 310
17.10 Exercisesp. 313
Chapter 18 Curvesp. 315
18.1 Application: Parametric Curvesp. 316
18.2 Properties of Bezier Curvesp. 321
18.3 The Matrix Formp. 323
18.4 Derivativesp. 324
18.5 Composite Curvesp. 326
18.6 The Geometry of Planar Curvesp. 327
18.7 Moving along a Curvep. 329
18.8 Exercisesp. 331
Appendix A PostScript Tutorialp. 333
A.1 A Warm-Up Examplep. 333
A.2 Overviewp. 336
A.3 Affine Mapsp. 338
A.4 Variablesp. 339
A.5 Loopsp. 340
A.6 CTMp. 341
Appendix B Selected Problem Solutionsp. 343
Glossaryp. 367
Bibliographyp. 371
Indexp. 373
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