Practical Linear Algebra

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**Descartes' Discovery** Local and Global
Coordinates: 2D

Going from Global to Local

Local and Global Coordinates: 3D

Stepping Outside the Box

Application: Creating Coordinates

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 IntersectionsChanging 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 Rules2 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 ViewMoving Things Around: Affine Maps in 2D

Coordinate Transformations

Affine and Linear Maps

Translations

More General Affine Maps

Mapping Triangles to Triangles

Composing Affine MapsEigen 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 MatricesAffine Maps in 3D

Affine Maps

Translations

Mapping Tetrahedra

Parallel Projections

Homogeneous Coordinates and Perspective MapsInteractions 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: Gauss-Jacobi and Gauss-SeidelGeneral 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

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 Bezier 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

**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,

"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."

-Joern 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

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