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# Linear Algebra Done Right

Sheldon Axler

• 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
• Exercises
• CHAPTER 7 Operators on Inner-Product Spaces
• 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