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A First Course in Linear Algebra

• Preface
• Dedication and Acknowledgements

Systems of Linear Equations

• What is Linear Algebra?
• Solving Systems of Linear Equations
• Reduced Row-Echelon Form
• Types of Solution Sets
• Homogeneous Systems of Equations
• Nonsingular Matrices

Vectors

• Vector Operations
• Linear Combinations
• Spanning Sets
• Linear Independence
• Linear Dependence and Spans
• Orthogonality

Matrices

• Matrix Operations
• Matrix Multiplication
• Matrix Inverses and Systems of Linear Equations
• Matrix Inverses and Nonsingular Matrices
• Column and Row Spaces
• Four Subsets

Vector Spaces

• Vector Spaces
• Subspaces
• Linear Independence and Spanning Sets
• Bases
• Dimension
• Properties of Dimension

Determinants

• Determinant of a Matrix
• Properties of Determinants of Matrices

Eigenvalues

• Eigenvalues and Eigenvectors
• Properties of Eigenvalues and Eigenvectors
• Similarity and Diagonalization

Linear Transformations

• Linear Transformations
• Injective Linear Transformations
• Surjective Linear Transformations
• Invertible Linear Transformations

Representations

• Vector Representations
• Matrix Representations
• Change of Basis
• Orthonormal Diagonalization

Preliminaries

• Complex Number Operations
• Sets

Archetypes

•   ABCDEFGHIJKLM
•   NOPQRSTUVWX

Reference

• Notation
• Definitions
• Theorems
• Diagrams
• Examples
• Sage
• Proof Techniques
• GFDL License