This book provides a thorough introduction to Linear Algebra suitable for a first course. It can be used in a course with a theoretical flavor or one with emphasis on applications.

- Systems of Linear Equations
- Systems of Equations and Matrices
- Gauss-Jordan Elimination
- Gaussian Elimination
- Applications: Systems of Equations in Everyday Life, Kirkhoff's Laws, Balancing Chemical Equations, Leontief's Model

- Matrices
- Matrix Arithmetic
- The properties of Matrix Arithmetic
- Row Multiplication and Matrix Multiplication; Finding the Inverse of a Matrix
- The System AX=B When A is Invertible
- The System AX=B When A is Not Invertible
- Applications: The Adjacency Matrix of a Graph, Markov Chains

- 1. Determinants
- Definition of the Determinant
- Expansion by Minors
- The Determinant and Elementary Row Operations
- The Determinant and Invertibility
- A Formula for A Inverse and Cramer's Rule

- 2. Two and Three Dimensional Euclidean Space
- The Definition of Euclidean Space, Vector Arithmetic
- The Properties of Vector Arithmetic, Collinearity, the Norm of a Vector
- The Dot Product, Projections
- Bases
- Orthonormal Bases
- The Cross Product
- Applications: Equations of Lines and Planes in Three-Dimensional Space

- 3. Vector Spaces
- The Definition of a Vector Space; Examples
- Subspaces
- The Linear Span of a Set of Vectors
- Bases and Linear Independence
- More on Bases and Linear Independence
- The Solution Space of a Homogeneous System
- The Row and Column Space of a Matrix
- Coordinate Matrices
- The Change of Basis Problem for Vectors

- 4. Linear Transformations
- Matrix Transformations
- Linear Transformations
- Some Simple Properties of Linear Transformations
- The Vector Space Lin(V,W); The Composition of Linear Transformations
- The Kernel and Range of a Linear Transformation
- Isomorphisms
- Linear Transformations from Rn to Rm
- The Matrix of a Linear Transformation
- The Change of Basis Problem for Linear Operators
- Applications: Plane Linear Transformations

- 5. Eigenvalues and Eigenvectors
- Eignevalues, Eignevectors and the Diagonalization of Matrices
- Computing the Eigenvalues and Eigenvectors of a Matrix
- More on the Diagonalization of Matrices
- The Diagonalizability of Linear Operators
- Applications: The Powers of a Diagonalizable Matrix and Recurrence Relations

- 6. Inner Product Spaces
- The Definition of an Inner Product Space
- Properties of the Inner Product
- Orthonormal Bases
- The Gram-Schmidt Process
- Orthogonal Diagonalizability

- 7. Applications
- Diffferential Equations
- Quadratic Forms
- Eigenfunctions and Orthogonal Sequences

- 8. Computational Methods
- Introduction; Floating Point Numbers and Roundoff Error
- Gaussian Elimination with parital Pivoting
- Conditioned Matrices
- Iterative Methods for Solving Systems of Equations
- Approximating Eigenvalues and Eigenvectors

- Appendix
- The Phrase "If and Only If"
- Functions

**Second edition, 1988, ISBN 0-15-542736-9, 504 pp., Hardcover**

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