## With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

No credit card required

## 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.

1. Preliminaries
2. Dedication
3. Preface
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
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
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
6. 3.6 Distance of a Point to a Line
7. 3.7 The Foot of a Point
8. 3.8 A Meeting Place: Computing Intersections
9. 3.9 Exercises
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
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
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
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
7. 7.7 Repeating Maps
8. 7.8 Exercises
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
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
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
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
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
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
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
18. Chapter 15: Eigen Things Revisited
1. 15.1 The Basics Revisited
2. 15.2 The Power Method
4. 15.4 Eigenfunctions
5. 15.5 Exercises
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
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
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?
10. 18.10 Exercises
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
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
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
26. Bibliography