Chapter 1: Vectors and Matrices

    Section 1.1: Fundamental Operations with Vectors

    Section 1.2: The Dot Product

    Section 1.3: An Introduction to Proof Techniques

    Section 1.4: Fundamental Operations with Matrices

    Section 1.5: Matrix Multiplication

Chapter 2: Systems of Linear Equations

    Section 2.1: Solving Linear Systems Using Gaussian Elimination

    Section 2.2: Gauss-Jordan Row Reduction and Reduced Row Echelon Form

    Section 2.3: Equivalent Systems, Rank, and Row Space

    Section 2.4: Inverses of Matrices

Chapter 3: Determinants and Eigenvalues

    Section 3.1: Introduction to Determinants

    Section 3.2: Determinants and Row Reduction

    Section 3.3: Further Properties of the Determinant

    Section 3.4: Eigenvalues and Diagonalization

    Summary of Techniques

Chapter 4: Finite Dimensional Vector Spaces

    Section 4.1: Introduction to Vector Spaces

    Section 4.2: Subspaces

    Section 4.3: Span

    Section 4.4: Linear Independence

    Section 4.5: Basis and Dimension

    Section 4.6: Constructing Special Bases

    Section 4.7: Coordinatization

Chapter 5: Linear Transformations

    Section 5.1: Introduction to Linear Transformations

    Section 5.2: The Matrix of a Linear Transformation

    Section 5.3: The Dimension Theorem

    Section 5.4: Isomorphism

    Section 5.5: Diagonalization of Linear Operators

Chapter 6: Orthogonality

    Section 6.1: Orthogonal Bases and the Gram-Schmidt Process

    Section 6.2: Orthogonal Complements

    Section 6.3: Orthogonal Diagonalization

Chapter 7: Complex Vector Spaces and General Inner Products

    Section 7.1: Complex n-Vectors and Matrices

    Section 7.2: Complex Eigenvalues and Eigenvectors

    Section 7.3: Complex Vector Spaces

    Section 7.4: Orthogonality in Cⁿ

    Section 7.5: Inner Product Spaces

Chapter 8: Additional Applications

    Section 8.1: Graph Theory

    Section 8.2: Ohm's Law

    Section 8.3: Least-Squares Polynomials

    Section 8.4: Markov Chains

    Section 8.5: Hill Substitution: An Introduction to Coding Theory

    Section 8.6: Change of Variables and the Jacobian

    Section 8.7: Rotation of Axes

    Section 8.8: Computer Graphics

    Section 8.9: Differential Equations

    Section 8.10: Least-Squares Solutions for Inconsistent Systems

    Section 8.11: Max-Min Problems in Rⁿ and the Hessian Matrix

Chapter 9: Numerical Methods

    Section 9.1: Numerical Methods for Solving Systems

    Section 9.2: LDU Decomposition

    Section 9.3: The Power Method for Finding Eigenvalues

Chapter 10: Further Horizons

    Section 10.1: Elementary Matrices

    Section 10.2: Function Spaces

    Section 10.3: Quadratic Forms

Appendix A: Miscellaneous Proofs

Appendix B: Functions

Appendix C: Complex Numbers

Appendix D: Computers and Calculators

Appendix E: Answers to Selected Exercises

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