Principles of Linear Algebra with Mathematica^®

Table of Contents

• Preface
• Conventions and Notations
• 1 An Introduction to Mathematica
  • 1.1 The Very Basics
  • 1.2 Basic Arithmetic
  • 1.3 Lists and Matrices
  • 1.4 Expressions versus Functions
  • 1.5 Plotting and Animations
  • 1.6 Solving Systems of Equations
  • 1.7 Basic Programming
• 2 Linear Systems of Equations and Matrices
  • 2.1 Linear Systems of Equations
  • 2.2 Augmented Matrix of a Linear System and Row Operations
  • 2.3 Some Matrix Arithmetic
• 3 Gauss-Jordan Elimination and Reduced Row Echelon Form
  • 3.1 Gauss-Jordan Elimination and rref
  • 3.2 Elementary Matrices
  • 3.3 Sensitivity of Solutions to Error in the Linear System
• 4 Applications of Linear Systems and Matrices
  • 4.1 Applications of Linear Systems to Geometry
  • 4.2 Applications of Linear Systems to Curve Fitting
  • 4.3 Applications of Linear Systems to Economics
  • 4.4 Applications of Matrix Multiplication to Geometry
  • 4.5 An Application of Matrix Multiplication to Economics
• 5 Determinants, Inverses, and Cramer's Rule
  • 5.1 Determinants and Inverses from the Adjoint Formula
  • 5.2 Finding Determinants by Expanding along Any Row or Column
  • 5.3 Determinants Found by Triangularizing Matrices
  • 5.4 LU Factorization
  • 5.5 Inverses from rref
  • 5.6 Cramer's Rule
• 6 Basic Vector Algebra Topics
  • 6.1 Vectors
  • 6.2 Dot Product
  • 6.3 Cross Product
  • 6.4 Vector Projection
• 7 A Few Advanced Vector Algebra Topics
  • 7.1 Rotations in Space
  • 7.2 "Rolling" a Circle along a Curve
  • 7.3 The TNB Frame
• 8 Independence, Basis, and Dimension for Subspaces of ℝⁿ
  • 8.1 Subspaces of ℝⁿ
  • 8.2 Independent and Dependent Sets of Vectors in ℝⁿ
  • 8.3 Basis and Dimension for Subspaces of ℝⁿ
  • 8.4 Vector Projection onto a Subspace of ℝⁿ
  • 8.5 The Gram-Schmidt Orthonormalization Process
• 9 Linear Maps from ℝⁿ to ℝ^m
  • 9.1 Basics about Linear Maps
  • 9.2 The Kernel and Image Subspaces of a Linear Map
  • 9.3 Composites of Two Linear Maps and Inverses
  • 9.4 Change of Bases for the Matrix Representation of a Linear Map
• 10 The Geometry of Linear and Affine Maps
  • 10.1 The Effect of a Linear Map on Area and Arclength in Two Dimensions
  • 10.2 The Decomposition of Linear Maps into Rotations, Reflections, and Rescalings in ℝ²
  • 10.3 The Effect of Linear Maps on Volume, Area, and Arclength in ℝ³
  • 10.4 Rotations, Reflections, and Rescalings in Three Dimensions
  • 10.5 Affine Maps
• 11 Least-Squares Fits and Pseudoinverses
  • 11.1 Pseudoinverse to a Nonsquare Matrix and Almost Solving an Overdetermined Linear System
  • 11.2 Fits and Pseudoinverses
  • 11.3 Least-Squares Fits and Pseudoinverses
• 12 Eigenvalues and Eigenvectors
  • 12.1 What Are Eigenvalues and Eigenvectors, and Why Do We Need Them?
  • 12.2 Summary of Definitions and Methods for Computing Eigenvalues and Eigenvectors as Well as the Exponential of a Matrix
  • 12.3 Applications of the Diagonalizability of Square Matrices
  • 12.4 Solving a Square First-Order Linear System of Differential Equations
  • 12.5 Basic Facts About Eigenvalues and Eigenvectors, and Diagonalizability
  • 12.6 The Geometry of the Ellipse Using Eigenvalues and Eigenvectors
  • 12.7 A Mathematica Eigen-Function
• Bibliography
• Indexes
  • Keyword Index
  • Index of Mathematica Commands