Elementary Linear Algebra develops and explains in careful detail the computational techniques and fundamental theoretical results central to a first course in linear algebra. This highly acclaimed text focuses on developing the abstract thinking essential for further mathematical study. The authors give early, intensive attention to the skills necessary to make students comfortable with mathematical proofs.

The text builds a gradual and smooth transition from computational results to general theory of abstract vector spaces. It also provides flexbile coverage of practical applications, exploring a comprehensive range of topics.

includes a wide variety of applications, technology tips and exercises, organized in chart format for easy reference. More than 310 numbered examples in the text at least one for each new concept or application.

Exercise sets ordered by increasing difficulty, many with multiple parts for a total of more than 2135 questions. Provides an early introduction to eigenvalues/eigenvectors. A Student solutions manual, containing fully worked out solutions and instructors manual available.

**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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