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Practical Linear Algebra
A Geometry Toolbox, Third Edition
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Book Description
Through many examples and realworld applications, Practical Linear Algebra: A Geometry Toolbox, Third Edition teaches undergraduatelevel linear algebra in a comprehensive, geometric, and algorithmic way. Designed for a onesemester 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 eigenanalysis, 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 twodimensional setting and then revisit these concepts and others in a threedimensional setting. They also discuss higher dimensions in various reallife applications. Triangles, polygons, conics, and curves are introduced as central applications of linear algebra.
Instead of using the standard theoremproof 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
Descartes’ Discovery
Local and Global Coordinates: 2D
Going from Global to Local
Local and Global Coordinates: 3D
Stepping Outside the Box
Application: Creating Coordinates
Here and There: Points and Vectors in 2D
Points and Vectors
What’s the Difference?
Vector Fields
Length of a Vector
Combining Points
Independence
Dot Product
Orthogonal Projections
Inequalities
Lining Up: 2D Lines
Defining a Line
Parametric Equation of a Line
Implicit Equation of a Line
Explicit Equation of a Line
Converting Between Parametric and Implicit Equations
Distance of a Point to a Line
The Foot of a Point
A Meeting Place: Computing Intersections
Changing Shapes: Linear Maps in 2D
Skew Target Boxes
The Matrix Form
Linear Spaces
Scalings
Reflections
Rotations
Shears
Projections
Areas and Linear Maps: Determinants
Composing Linear Maps
More on Matrix Multiplication
Matrix Arithmetic Rules
2 x 2 Linear Systems
Skew Target Boxes Revisited
The Matrix Form
A Direct Approach: Cramer’s Rule
Gauss Elimination
Pivoting
Unsolvable Systems
Underdetermined Systems
Homogeneous Systems
Undoing Maps: Inverse Matrices
Defining a Map
A Dual View
Moving Things Around: Affine Maps in 2D
Coordinate Transformations
Affine and Linear Maps
Translations
More General Affine Maps
Mapping Triangles to Triangles
Composing Affine Maps
Eigen Things
Fixed Directions
Eigenvalues
Eigenvectors
Striving for More Generality
The Geometry of Symmetric Matrices
Quadratic Forms
Repeating Maps
3D Geometry
From 2D to 3D
Cross Product
Lines
Planes
Scalar Triple Product
Application: Lighting and Shading
Linear Maps in 3D
Matrices and Linear Maps
Linear Spaces
Scalings
Reflections
Shears
Rotations
Projections
Volumes and Linear Maps: Determinants
Combining Linear Maps
Inverse Matrices
More on Matrices
Affine Maps in 3D
Affine Maps
Translations
Mapping Tetrahedra
Parallel Projections
Homogeneous Coordinates and Perspective Maps
Interactions in 3D
Distance between a Point and a Plane
Distance between Two Lines
Lines and Planes: Intersections
Intersecting a Triangle and a Line
Reflections
Intersecting Three Planes
Intersecting Two Planes
Creating Orthonormal Coordinate Systems
Gauss for Linear Systems
The Problem
The Solution via Gauss Elimination
Homogeneous Linear Systems
Inverse Matrices
LU Decomposition
Determinants
Least Squares
Application: Fitting Data to a Femoral Head
Alternative System Solvers
The Householder Method
Vector Norms
Matrix Norms
The Condition Number
Vector Sequences
Iterative System Solvers: GaussJacobi and GaussSeidel
General Linear Spaces
Basic Properties of Linear Spaces
Linear Maps
Inner Products
GramSchmidt Orthonormalization
A Gallery of Spaces
Eigen Things Revisited
The Basics Revisited
The Power Method
Application: Google Eigenvector
Eigenfunctions
The Singular Value Decomposition
The Geometry of the 2 x 2 Case
The General Case
SVD Steps
Singular Values and Volumes
The Pseudoinverse
Least Squares
Application: Image Compression
Principal Components Analysis
Breaking It Up: Triangles
Barycentric Coordinates
Affine Invariance
Some Special Points
2D Triangulations
A Data Structure
Application: Point Location
3D Triangulations
Putting Lines Together: Polylines and Polygons
Polylines
Polygons
Convexity
Types of Polygons
Unusual Polygons
Turning Angles and Winding Numbers
Area
Application: Planarity Test
Application: Inside or Outside?
Conics
The General Conic
Analyzing Conics
General Conic to Standard Position
Curves
Parametric Curves
Properties of Bézier Curves
The Matrix Form
Derivatives
Composite Curves
The Geometry of Planar Curves
Moving along a Curve
Appendix A: Glossary
Appendix B: Selected Exercise Solutions
Bibliography
Index
Exercises appear at the end of each chapter.
Reviews
Praise for the Second Edition:
"… 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. … Recommended."
—J.R. Burke, CHOICE, September 2005"I picked up this book with the thought, ‘oh, another linear algebra text.’ I was pleasantly surprised, upon examination, that it is not just another one. The standard linear algebra material is presented with good motivating stories, illustrations, and examples."
—CMS Notes, February 2006"[The] mixture of linear algebra, geometry, and numerical aspects is very interesting and will probably stimulate the students."
—Bulletin of the Belgian Mathematical Society, December 2005"Teaching computer graphics and mathematics in the study program Digital Media, I have found Practical Linear Algebra to be precisely the kind of book I’ve long been looking for. It covers all topics that are vital for computer graphics, even gives lots of applications in that field, including for instance PostScript, and does so in a very practical but nonetheless rigorous manner. I find many elements of my own teaching in this book, including the handmade style of drawings (rendering them more ‘handson’), using a geometric shape to illustrate the action of a 2x2 matrix, and deriving the determinant from the computation of areas and volumes instead of plainly presenting a formula. I have recommended this book strongly to all students in my firstyear courses and will continue to do so."
—Jörn Loviscach, University of Applied Sciences Bremen"I was impressed with the applications, especially those related to computer graphics. … I think some faculty will be interested in using the book because the geometric descriptions and applications are very nice."
—Linda Patton, Cal Poly San Luis Obispo"I just purchased your book … and I immediately fell in love with it. I love the nice illustrations and diagrams, which are very helpful in promoting an intuitive understanding of every concept. I am a clinician investigator at the National Institutes of Health who is conducting MRI research of the heart. My main interest is finite deformation of the heart muscle structures; however, since I do not have an engineering background, I have been looking for a nice textbook on linear algebra."
—Hiroshi Ashikaga, National Institutes of Health"After having finished your book today, I agree completely with the very favorable reviews that I saw on the Internet. It was an enlightening experience to brush up on my old university math with this book. The ‘sketchy’ way of explaining things sure worked for me."
—Anneke SichererRoetman, Maritime Research Institute Netherlands"… it’s done a world of wonder for me, as I need to review my linear algebra to prepare for studying computer graphics. I really can’t thank you enough."
—Daniel Kurtz, Northeastern University"Practical Linear Algebra is great. I write software for the image analysis of medical images where in several occasions I have had to deal with eigen things and [others]. I have been using your book as a valuable reference to refresh and understand these concepts that I studied when I was a student."
—Diego Bordegari
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