Introduction to Linear Algebra with Applications

Published by Saunders College Publishing

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.

Contents

• 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