You are previewing Practical Linear Algebra, 3rd Edition.
O'Reilly logo
Practical Linear Algebra, 3rd Edition

Book Description

Through many examples and real-world applications, Practical Linear Algebra: A Geometry Toolbox, Third Edition teaches undergraduate-level linear algebra in a comprehensive, geometric, and algorithmic way. Designed for a one-semester linear algebra course at the undergraduate level, the book gives instructors the option of tailoring the course for the primary interests: math, engineering, science, computer graphics, and geometric modeling.

New to the Third Edition

  • More exercises and applications
  • Coverage of singular value decomposition and its application to the pseudoinverse, principal components analysis, and image compression
  • More attention to eigen-analysis, including eigenfunctions and the Google matrix
  • Greater emphasis on orthogonal projections and matrix decompositions, which are tied to repeated themes such as the concept of least squares

To help students better visualize and understand the material, the authors introduce the fundamental concepts of linear algebra first in a two-dimensional setting and then revisit these concepts and others in a three-dimensional setting. They also discuss higher dimensions in various real-life applications. Triangles, polygons, conics, and curves are introduced as central applications of linear algebra.

Instead of using the standard theorem-proof approach, the text presents many examples and instructional illustrations to help students develop a robust, intuitive understanding of the underlying concepts. The authors’ website also offers the illustrations for download and includes Mathematica® code and other ancillary materials.

Table of Contents

  1. Preliminaries
  2. Dedication
  3. Preface
    1. Review of Contents
    2. Classroom Use
    3. Website
  4. Chapter 1: Descartes’ Discovery
    1. 1.1 Local and Global Coordinates: 2D
    2. 1.2 Going from Global to Local
    3. 1.3 Local and Global Coordinates: 3D
    4. 1.4 Stepping Outside the Box
    5. 1.5 Application: Creating Coordinates
    6. 1.6 Exercises
      1. Figure 1.1
      2. Figure 1.2
      3. Figure 1.3
      4. Figure 1.4
      1. Sketch 1.1
      2. Sketch 1.2
      3. Sketch 1.3
      4. Sketch 1.4
      5. Sketch 1.5
      6. Sketch 1.6
      7. Sketch 1.7
      8. Sketch 1.8
      9. Sketch 1.9
  5. Chapter 2: Here and There: Points and Vectors in 2D
    1. 2.1 Points and Vectors
    2. 2.2 What’s the Difference?
    3. 2.3 Vector Fields
    4. 2.4 Length of a Vector
    5. 2.5 Combining Points
    6. 2.6 Independence
    7. 2.7 Dot Product
    8. 2.8 Orthogonal Projections
    9. 2.9 Inequalities
    10. 2.10 Exercises
      1. Figure 2.1
      2. Figure 2.2
      3. Figure 2.3
      4. Figure 2.4
      5. Figure 2.5
      1. Sketch 2.1
      2. Sketch 2.2
      3. Sketch 2.3
      4. Sketch 2.4
      5. Sketch 2.5
      6. Sketch 2.6
      7. Sketch 2.7
      8. Sketch 2.8
      9. Sketch 2.9
      10. Sketch 2.10
      11. Sketch 2.11
      12. Sketch 2.12
      13. Sketch 2.13
      14. Sketch 2.14
      15. Sketch 2.15
      16. Sketch 2.16
      17. Sketch 2.17
      18. Sketch 2.18
      19. Sketch 2.19
      20. Sketch 2.20
      21. Sketch 2.21
      22. Sketch 2.22
  6. Chapter 3: Lining Up: 2D Lines
    1. 3.1 Defining a Line
    2. 3.2 Parametric Equation of a Line
    3. 3.3 Implicit Equation of a Line
    4. 3.4 Explicit Equation of a Line
    5. 3.5 Converting Between Parametric and Implicit Equations
      1. 3.5.1 Parametric to Implicit
      2. 3.5.2 Implicit to Parametric
    6. 3.6 Distance of a Point to a Line
      1. 3.6.1 Starting with an Implicit Line
      2. 3.6.2 Starting with a Parametric Line
    7. 3.7 The Foot of a Point
    8. 3.8 A Meeting Place: Computing Intersections
      1. 3.8.1 Parametric and Implicit
      2. 3.8.2 Both Parametric
      3. 3.8.3 Both Implicit
    9. 3.9 Exercises
      1. Figure 3.1
      2. Figure 3.2
      3. Figure 3.3
      4. Figure 3.4
      1. Sketch 3.1
      2. Sketch 3.2
      3. Sketch 3.3
      4. Sketch 3.4
      5. Sketch 3.5
      6. Sketch 3.6
      7. Sketch 3.7
      8. Sketch 3.8
      9. Sketch 3.9
      10. Sketch 3.10
      11. Sketch 3.11
  7. Chapter 4: Changing Shapes: Linear Maps in 2D
    1. 4.1 Skew Target Boxes
    2. 4.2 The Matrix Form
    3. 4.3 Linear Spaces
    4. 4.4 Scalings
    5. 4.5 Reflections
    6. 4.6 Rotations
    7. 4.7 Shears
    8. 4.8 Projections
    9. 4.9 Areas and Linear Maps: Determinants
    10. 4.10 Composing Linear Maps
    11. 4.11 More on Matrix Multiplication
    12. 4.12 Matrix Arithmetic Rules
    13. 4.13 Exercises
      1. Figure 4.1
      2. Figure 4.2
      3. Figure 4.3
      4. Figure 4.4
      5. Figure 4.5
      6. Figure 4.6
      7. Figure 4.7
      8. Figure 4.8
      9. Figure 4.9
      10. Figure 4.10
      11. Figure 4.11
      12. Figure 4.12
      1. Sketch 4.1
      2. Sketch 4.2
      3. Sketch 4.3
      4. Sketch 4.4
      5. Sketch 4.5
      6. Sketch 4.6
      7. Sketch 4.7
      8. Sketch 4.8
      9. Sketch 4.9
      10. Sketch 4.10
      11. Sketch 4.11
      12. Sketch 4.12
  8. Chapter 5: 2 × 2 Linear Systems
    1. 5.1 Skew Target Boxes Revisited
    2. 5.2 The Matrix Form
    3. 5.3 A Direct Approach: Cramer’s Rule
    4. 5.4 Gauss Elimination
    5. 5.5 Pivoting
    6. 5.6 Unsolvable Systems
    7. 5.7 Underdetermined Systems
    8. 5.8 Homogeneous Systems
    9. 5.9 Undoing Maps: Inverse Matrices
    10. 5.10 Defining a Map
    11. 5.11 A Dual View
    12. 5.12 Exercises
      1. Figure 5.1
      2. Figure 5.2
      3. Figure 5.3
      4. Figure 5.4
      5. Figure 5.5
      6. Figure 5.6
      1. Sketch 5.1
      2. Sketch 5.2
      3. Sketch 5.3
      4. Sketch 5.4
      5. Sketch 5.5
      6. Sketch 5.6
      7. Sketch 5.7
      8. Sketch 5.8
      9. Sketch 5.9
      10. Sketch 5.10
  9. Chapter 6: Moving Things Around: Affine Maps in 2D
    1. 6.1 Coordinate Transformations
    2. 6.2 Affine and Linear Maps
    3. 6.3 Translations
    4. 6.4 More General Affine Maps
    5. 6.5 Mapping Triangles to Triangles
    6. 6.6 Composing Affine Maps
    7. 6.7 Exercises
      1. Figure 6.1
      2. Figure 6.2
      3. Figure 6.3
      4. Figure 6.4
      5. Figure 6.5
      6. Figure 6.6
      7. Figure 6.7
      8. Figure 6.8
      9. Figure 6.9
      1. Sketch 6.1
      2. Sketch 6.2
      3. Sketch 6.3
      4. Sketch 6.4
      5. Sketch 6.5
      6. Sketch 6.6
      7. Sketch 6.7
      8. Sketch 6.8
  10. Chapter 7: Eigen Things
    1. 7.1 Fixed Directions
    2. 7.2 Eigenvalues
    3. 7.3 Eigenvectors
    4. 7.4 Striving for More Generality
    5. 7.5 The Geometry of Symmetric Matrices
    6. 7.6 Quadratic Forms
    7. 7.7 Repeating Maps
    8. 7.8 Exercises
      1. Figure 7.1
      2. Figure 7.2
      3. Figure 7.3
      4. Figure 7.4
      5. Figure 7.5
      6. Figure 7.6
      7. Figure 7.7
      8. Figure 7.8
      9. Figure 7.9
      10. Figure 7.10
      11. Figure 7.11
  11. Chapter 8: 3D Geometry
    1. 8.1 From 2D to 3D
    2. 8.2 Cross Product
    3. 8.3 Lines
    4. 8.4 Planes
    5. 8.5 Scalar Triple Product
    6. 8.6 Application: Lighting and Shading
    7. 8.7 Exercises
      1. Figure 8.1
      2. Figure 8.2
      3. Figure 8.3
      4. Figure 8.4
      1. Sketch 8.1
      2. Sketch 8.2
      3. Sketch 8.3
      4. Sketch 8.4
      5. Sketch 8.5
      6. Sketch 8.6
      7. Sketch 8.7
      8. Sketch 8.8
      9. Sketch 8.9
      10. Sketch 8.10
      11. Sketch 8.11
      12. Sketch 8.12
      13. Sketch 8.13
      14. Sketch 8.14
  12. Chapter 9: Linear Maps in 3D
    1. 9.1 Matrices and Linear Maps
    2. 9.2 Linear Spaces
    3. 9.3 Scalings
    4. 9.4 Reflections
    5. 9.5 Shears
    6. 9.6 Rotations
    7. 9.7 Projections
    8. 9.8 Volumes and Linear Maps: Determinants
    9. 9.9 Combining Linear Maps
    10. 9.10 Inverse Matrices
    11. 9.11 More on Matrices
    12. 9.12 Exercises
      1. Figure 9.1
      2. Figure 9.2
      3. Figure 9.3
      4. Figure 9.4
      5. Figure 9.5
      6. Figure 9.6
      7. Figure 9.7
      8. Figure 9.8
      1. Sketch 9.1
      2. Sketch 9.2
      3. Sketch 9.3
      4. Sketch 9.4
      5. Sketch 9.5
      6. Sketch 9.6
      7. Sketch 9.7
      8. Sketch 9.8
      9. Sketch 9.9
      10. Sketch 9.10
  13. Chapter 10: Affine Maps in 3D
    1. 10.1 Affine Maps
    2. 10.2 Translations
    3. 10.3 Mapping Tetrahedra
    4. 10.4 Parallel Projections
    5. 10.5 Homogeneous Coordinates and Perspective Maps
    6. 10.6 Exercises
      1. Figure 10.1
      2. Figure 10.2
      3. Figure 10.3
      4. Figure 10.4
      5. Figure 10.5
      6. Figure 10.6
      7. Figure 10.7
      1. Sketch 10.1
      2. Sketch 10.2
      3. Sketch 10.3
      4. Sketch 10.4
      5. Sketch 10.5
      6. Sketch 10.6
      7. Sketch 10.7
      8. Sketch 10.8
      9. Sketch 10.9
  14. Chapter 11: Interactions in 3D
    1. 11.1 Distance Between a Point and a Plane
    2. 11.2 Distance Between Two Lines
    3. 11.3 Lines and Planes: Intersections
    4. 11.4 Intersecting a Triangle and a Line
    5. 11.5 Reflections
    6. 11.6 Intersecting Three Planes
    7. 11.7 Intersecting Two Planes
    8. 11.8 Creating Orthonormal Coordinate Systems
    9. 11.9 Exercises
      1. Figure 11.1
      1. Sketch 11.1
      2. Sketch 11.2
      3. Sketch 11.3
      4. Sketch 11.4
      5. Sketch 11.5
      6. Sketch 11.6
      7. Sketch 11.7
      8. Sketch 11.8
      9. Sketch 11.9
      10. Sketch 11.10
      11. Sketch 11.11
      12. Sketch 11.12
      13. Sketch 11.13
      14. Sketch 11.14
      15. Sketch 11.15
      16. Sketch 11.16
  15. Chapter 12: Gauss for Linear Systems
    1. 12.1 The Problem
    2. 12.2 The Solution via Gauss Elimination
    3. 12.3 Homogeneous Linear Systems
    4. 12.4 Inverse Matrices
    5. 12.5 LU Decomposition
    6. 12.6 Determinants
    7. 12.7 Least Squares
    8. 12.8 Application: Fitting Data to a Femoral Head
    9. 12.9 Exercises
      1. Figure 12.1
      2. Figure 12.2
      3. Figure 12.3
      4. Figure 12.4
      5. Figure 12.5
      1. Sketch 12.1
      2. Sketch 12.2
      3. Sketch 12.3
  16. Chapter 13: Alternative System Solvers
    1. 13.1 The Householder Method
    2. 13.2 Vector Norms
    3. 13.3 Matrix Norms
    4. 13.4 The Condition Number
    5. 13.5 Vector Sequences
    6. 13.6 Iterative System Solvers: Gauss-Jacobi and Gauss-Seidel
    7. 13.7 Exercises
      1. Figure 13.1
      2. Figure 13.2
      3. Figure 13.3
      4. Figure 13.4
      5. Figure 13.5
      1. Sketch 13.1
  17. Chapter 14: General Linear Spaces
    1. 14.1 Basic Properties of Linear Spaces
    2. 14.2 Linear Maps
    3. 14.3 Inner Products
    4. 14.4 Gram-Schmidt Orthonormalization
    5. 14.5 A Gallery of Spaces
    6. 14.6 Exercises
      1. Figure 14.1
      2. Figure 14.2
      3. Figure 14.3
      1. Sketch 14.1
  18. Chapter 15: Eigen Things Revisited
    1. 15.1 The Basics Revisited
    2. 15.2 The Power Method
    3. 15.3 Application: Google Eigenvector
    4. 15.4 Eigenfunctions
    5. 15.5 Exercises
      1. Figure 15.1
      2. Figure 15.2
      3. Figure 15.3
      4. Figure 15.4
  19. Chapter 16: The Singular Value Decomposition
    1. 16.1 The Geometry of the 2 × 2 Case
    2. 16.2 The General Case
    3. 16.3 SVD Steps
    4. 16.4 Singular Values and Volumes
    5. 16.5 The Pseudoinverse
    6. 16.6 Least Squares
    7. 16.7 Application: Image Compression
    8. 16.8 Principal Components Analysis
    9. 16.9 Exercises
      1. Figure 16.1
      2. Figure 16.2
      3. Figure 16.3
      4. Figure 16.4
      5. Figure 16.5
      6. Figure 16.6
      7. Figure 16.7
      8. Figure 16.8
      9. Figure 16.9
      10. Figure 16.10
  20. Chapter 17: Breaking It Up: Triangles
    1. 17.1 Barycentric Coordinates
    2. 17.2 Affine Invariance
    3. 17.3 Some Special Points
    4. 17.4 2D Triangulations
    5. 17.5 A Data Structure
    6. 17.6 Application: Point Location
    7. 17.7 3D Triangulations
    8. 17.8 Exercises
      1. Figure 17.1
      2. Figure 17.2
      3. Figure 17.3
      1. Sketch 17.1
      2. Sketch 17.2
      3. Sketch 17.3
      4. Sketch 17.4
      5. Sketch 17.5
      6. Sketch 17.6
      7. Sketch 17.7
      8. Sketch 17.8
      9. Sketch 17.9
      10. Sketch 17.10
      11. Sketch 17.11
      12. Sketch 17.12
  21. Chapter 18: Putting Lines Together: Polylines and Polygons
    1. 18.1 Polylines
    2. 18.2 Polygons
    3. 18.3 Convexity
    4. 18.4 Types of Polygons
    5. 18.5 Unusual Polygons
    6. 18.6 Turning Angles and Winding Numbers
    7. 18.7 Area
    8. 18.8 Application: Planarity Test
    9. 18.9 Application: Inside or Outside?
      1. 18.9.1 Even-Odd Rule
      2. 18.9.2 Nonzero Winding Number
    10. 18.10 Exercises
      1. Figure 18.1
      2. Figure 18.2
      3. Figure 18.3
      4. Figure 18.4
      5. Figure 18.5
      6. Figure 18.6
      7. Figure 18.7
      1. Sketch 18.1
      2. Sketch 18.2
      3. Sketch 18.3
      4. Sketch 18.4
      5. Sketch 18.5
      6. Sketch 18.6
      7. Sketch 18.7
      8. Sketch 18.8
      9. Sketch 18.9
      10. Sketch 18.10
      11. Sketch 18.11
      12. Sketch 18.12
      13. Sketch 18.13
      14. Sketch 18.14
      15. Sketch 18.15
      16. Sketch 18.16
      17. Sketch 18.17
  22. Chapter 19: Conics
    1. 19.1 The General Conic
    2. 19.2 Analyzing Conics
    3. 19.3 General Conic to Standard Position
    4. 19.4 Exercises
      1. Figure 19.1
      2. Figure 19.2
      3. Figure 19.3
      1. Sketch 19.1
      2. Sketch 19.2
      3. Sketch 19.3
      4. Sketch 19.4
  23. Chapter 20: Curves
    1. 20.1 Parametric Curves
    2. 20.2 Properties of Bézier Curves
    3. 20.3 The Matrix Form
    4. 20.4 Derivatives
    5. 20.5 Composite Curves
    6. 20.6 The Geometry of Planar Curves
    7. 20.7 Moving along a Curve
    8. 20.8 Exercises
      1. Figure 20.1
      2. Figure 20.2
      3. Figure 20.3
      4. Figure 20.4
      5. Figure 20.5
      6. Figure 20.6
      7. Figure 20.7
      8. Figure 20.8
      9. Figure 20.9
      10. Figure 20.10
      11. Figure 20.11
      1. Sketch 20.1
      2. Sketch 20.2
      3. Sketch 20.3
      4. Sketch 20.4
      5. Sketch 20.5
      6. Sketch 20.6
      7. Sketch 20.7
      8. Sketch 20.8
      9. Sketch 20.9
  24. A: Glossary
  25. B: Selected Exercise Solutions
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
    13. Chapter 13
    14. Chapter 14
    15. Chapter 15
    16. Chapter 16
    17. Chapter 17
    18. Chapter 18
    19. Chapter 19
    20. Chapter 20
      1. Figure B.1.
      2. Figure B.2.
      3. Figure B.3.
  26. Bibliography