Linear Algebra and Its Applications

by David Caldwell Lay; Karen Guardino

• ISBN13: 9780201824780
• ISBN10: 0201824787
• Edition: 2nd
• Format: Hardcover
• Copyright: 1996-12-01
• Publisher: ADDISON WESLEY

Summary

Linear algebra is relatively easy for students during the early stages of the course, when the material is presented in a familiar, concrete setting. But when abstract concepts are introduced, students often hit a brick wall. Instructors seem to agree that certain concepts (such as linear independence, spanning, subspace, vector space, and linear transformations), are not easily understood, and require time to assimilate. Since they are fundamental to the study of linear algebra, students' understanding of these concepts is vital to their mastery of the subject. Lay introduces these concepts early in a familiar, concrete Rn setting, develops them gradually, and returns to them again and again throughout the text so that when discussed in the abstract, these concepts are more accessible.

Table of Contents

Linear Equations in Linear Algebra

Introductory Example: Linear Models in Economics and Engineering

Systems of Linear Equations

Row Reduction and Echelon Forms

Vector Equations

The Matrix Equation Ax = b

Solution Sets of Linear Systems

Applications of Linear Systems

Linear Independence

Introduction to Linear Transformations

The Matrix of a Linear Transformation

Linear Models in Business, Science, and Engineering

Supplementary Exercises

Matrix Algebra

Introductory Example: Computer Models in Aircraft Design

Matrix Operations

The Inverse of a Matrix

Characterizations of Invertible Matrices

Partitioned Matrices

Matrix Factorizations

The Leontief Input=Output Model

Applications to Computer Graphics

Subspaces of Rn

Dimension and Rank

Supplementary Exercises

Determinants

Introductory Example: Determinants in Analytic Geometry

Introduction to Determinants

Properties of Determinants

Cramer's Rule, Volume, and Linear Transformations

Supplementary Exercises

Vector Spaces

Introductory Example: Space Flight and Control Systems

Vector Spaces and Subspaces

Null Spaces, Column Spaces, and Linear Transformations

Linearly Independent Sets Bases

Coordinate Systems

The Dimension of a Vector Space

Rank

Change of Basis

Applications to Difference Equations

Applications to Markov Chains

Supplementary Exercises

Eigenvalues and Eigenvectors

Introductory Example: Dynamical Systems and Spotted Owls

Eigenvectors and Eigenvalues

The Characteristic Equation

Diagonalization

Eigenvectors and Linear Transformations

Complex Eigenvalues

Discrete Dynamical Systems

Applications to Differential Equations

Iterative Estimates for Eigenvalues

Supplementary Exercises

Orthogonality and Least Squares

Introductory Example: Readjusting the North American Datum

Inner Product, Length, and Orthogonality

Orthogonal Sets

Orthogonal Projections

The Gram-Schmidt Process

Least-Squares Problems

Applications to Linear Models

Inner Product Spaces

Applications of Inner Product Spaces

Supplementary Exercises

Symmetric Matrices and Quadratic Forms

Introductory Example: Multichannel Image Processing

Diagonalization of Symmetric Matrices

Quadratic Forms

Constrained Optimization

The Singular Value Decomposition

Applications to Image Processing and Statistics

Supplementary Exercises (ONLINE ONLY)

The Geometry of Vector Spaces

Introductory Example: The Platonic Solids

Affine Combinations

Affine Independence

Convex Combinations

Hyperplanes

Polytopes

Curves and Surfaces

Supplementary Exercises (ONLINE ONLY)

Optimization

Introductory Example: The Berlin Airlift

Matrix Games

Linear Programming - Geometric Method

Linear Programming - Simplex Method

Duality

Supplementary Exercises

Appendices

Uniqueness of the Reduced Echelon Form

Complex Numbers

Glossary

Answers to Odd-Numbered Exercises

Index

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