Applied Linear Algebra And Matrix Analysis

by Shores, Thomas S.

• ISBN13: 9780387331959
• ISBN10: 0387331956
• Format: Paperback
• Copyright: 2007-08-01
• Publisher: Springer Verlag

Summary

This text is intended for a one or two semester sophomore level course in linear algebra. It is designed to provide a balance of applications, theory and computation, and to emphasize their interdependence. The text has a strong orientation towards numerical computation and the linear algebra needed in applied mathematics. At the same time, it contains a rigorous and self-contained development of most of the traditional topics in a linear algebra course. It provides background for numerous projects, which frequently require computational tools, but is not tied to any one computational platform. A comprehensive set of exercises and projects is included.

Table of Contents

Linear Systems of Equations	p. 1
Some Examples	p. 1
Notation and a Review of Numbers	p. 9
Gaussian Elimination: Basic Ideas	p. 21
Gaussian Elimination: General Procedure	p. 33
Computational Notes and Projects	p. 46
Matrix Algebra	p. 55
Matrix Addition and Scalar Multiplication	p. 55
Matrix Multiplication	p. 62
Applications of Matrix Arithmetic	p. 71
Special Matrices and Transposes	p. 86
Matrix Inverses	p. 101
Basic Properties of Determinants	p. 114
Computational Notes and Projects	p. 129
Vector Spaces	p. 145
Definitions and Basic Concepts	p. 145
Subspaces	p. 161
Linear Combinations	p. 170
Subspaces Associated with Matrices and Operators	p. 183
Bases and Dimension	p. 191
Linear Systems Revisited	p. 198
Computational Notes and Projects	p. 208
Geometrical Aspects of Standard Spaces	p. 211
Standard Norm and Inner Product	p. 211
Applications of Norms and Inner Products	p. 221
Orthogonal and Unitary Matrices	p. 233
Change of Basis and Linear Operators	p. 242
Computational Notes and Projects	p. 247
The Eigenvalue Problem	p. 251
Definitions and Basic Properties	p. 251
Similarity and Diagonalization	p. 263
Applications to Discrete Dynamical Systems	p. 272
Orthogonal Diagonalization	p. 282
Schur Form and Applications	p. 287
The Singular Value Decomposition	p. 291
Computational Notes and Projects	p. 294
Geometrical Aspects of Abstract Spaces	p. 305
Normed Spaces	p. 305
Inner Product Spaces	p. 312
Gram-Schmidt Algorithm	p. 323
Linear Systems Revisited	p. 333
Operator Norms	p. 342
Computational Notes and Projects	p. 348
Table of Symbols	p. 355
Solutions to Selected Exercises	p. 357
References	p. 375
Index	p. 377
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