Practical linear algebra: a geometry toolbox

Abstract

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: 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 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 Exercises appear at the end of each chapter.