Practical Linear Algebra: A Geometry Toolbox, Third Edition, 3rd Edition (Hardback) book cover

Practical Linear Algebra

A Geometry Toolbox, Third Edition, 3rd Edition

By Gerald Farin, Dianne Hansford

A K Peters/CRC Press

514 pages | 492 B/W Illus.

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Hardback: 9781466579569
pub: 2013-08-19
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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.


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 hand-made style of drawings (rendering them more ‘hands-on’), 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 first-year 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 Sicherer-Roetman, 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

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


Dot Product

Orthogonal Projections


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






Areas and Linear Maps: Determinants

Composing Linear Maps

More on Matrix Multiplication

Matrix Arithmetic Rules

2 x2 Linear Systems

Skew Target Boxes Revisited

The Matrix Form

A Direct Approach: Cramer’s Rule

Gauss Elimination


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


More General Affine Maps

Mapping Triangles to Triangles

Composing Affine Maps

Eigen Things

Fixed Directions



Striving for More Generality

The Geometry of Symmetric Matrices

Quadratic Forms

Repeating Maps

3D Geometry

From 2D to 3D

Cross Product



Scalar Triple Product

Application: Lighting and Shading

Linear Maps in 3D

Matrices and Linear Maps

Linear Spaces






Volumes and Linear Maps: Determinants

Combining Linear Maps

Inverse Matrices

More on Matrices

Affine Maps in 3D

Affine Maps


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


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


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: Gauss-Jacobi and Gauss-Seidel

General Linear Spaces

Basic Properties of Linear Spaces

Linear Maps

Inner Products

Gram-Schmidt Orthonormalization

A Gallery of Spaces

Eigen Things Revisited

The Basics Revisited

The Power Method

Application: Google Eigenvector


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




Types of Polygons

Unusual Polygons

Turning Angles and Winding Numbers


Application: Planarity Test

Application: Inside or Outside?


The General Conic

Analyzing Conics

General Conic to Standard Position


Parametric Curves

Properties of Bézier Curves

The Matrix Form


Composite Curves

The Geometry of Planar Curves

Moving along a Curve

Appendix A: Glossary

Appendix B: Selected Exercise Solutions



Exercises appear at the end of each chapter.

Subject Categories

BISAC Subject Codes/Headings:
COMPUTERS / Computer Graphics
MATHEMATICS / Algebra / General