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Linear Algebra Done Right

Table of Contents

• Preface to the Instructor
• Preface to the Student
• Acknowledgments
• CHAPTER 1 Vector Spaces
  • Complex Numbers
  • Definition of Vector Space
  • Properties of Vector Spaces
  • Subspaces
  • Sums and Direct Sums
  • Exercises
• CHAPTER 2 Finite-Dimensional Vector Spaces
  • Span and Linear Independence
  • Bases
  • Dimension
  • Exercises
• CHAPTER 3 Linear Maps
  • Definitions and Examples
  • Null Spaces and Ranges
  • The Matrix of a Linear Map
  • Invertibility
  • Exercises
• CHAPTER 4 Polynomials
  • Degree
  • Complex Coefficients
  • Real Coefficients
  • Exercises
• CHAPTER 5 Eigenvalues and Eigenvectors
  • Invariant Subspaces
  • Polynomials Applied to Operators
  • Upper-Triangular Matrices
  • Diagonal Matrices
  • Invariant Subspaces on Real Vector Spaces
  • Exercises
• CHAPTER 6 Inner-Product Spaces
  • Inner Products
  • Norms
  • Orthonormal Bases
  • Orthogonal Projections and Minimization Problems
  • Linear Functionals and Adjoints
  • Exercises
• CHAPTER 7 Operators on Inner-Product Spaces
  • Self-Adjoint and Normal Operators
  • The Spectral Theorem
  • Normal Operators on Real Inner-Product Spaces
  • Positive Operators
  • Isometries
  • Polar and Singular-Value Decompositions
  • Exercises
• CHAPTER 8 Operators on Complex Vector Spaces
  • Generalized Eigenvectors
  • The Characteristic Polynomial
  • Decomposition of an Operator
  • Square Roots
  • The Minimal Polynomial
  • Jordan Form
  • Exercises
• CHAPTER 9 Operators on Real Vector Spaces
  • Eigenvalues of Square Matrices
  • Block Upper-Triangular Matrices
  • The Characteristic Polynomial
  • Exercises
• CHAPTER 10 Trace and Determinant
  • Change of Basis
  • Trace
  • Determinant of an Operator
  • Determinant of a Matrix
  • Volume
• Exercises
• Symbol Index
• Index

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