Elementary Linear Algebra with Applications

by Kolman, Bernard; Hill, David

• ISBN13: 9780130457875
• ISBN10: 0130457876
• Edition: 8th
• Format: Hardcover
• Copyright: 2008-01-01
• Publisher: Prentice Hall

Summary

This book presents the basic ideas of linear algebra in a manner that users will find understandable. It offers a fine balance between abstraction/theory and computational skills, and gives readers an excellent opportunity to learn how to handle abstract concepts.Included in this comprehensive and easy-to-follow manual are these topics: linear equations and matrices; solving linear systems; real vector spaces; inner product spaces; linear transformations and matrices; determinants; eigenvalues and eigenvectors; differential equations; and MATLAB for linear algebra.Because this book gives real applications for linear algebraic basic ideas and computational techniques, it is useful as a reference work for mathematicians and those in field of computer science.

Table of Contents

Preface	xi
Linear Equations and Matrices	1	(76)
Systems of Linear Equations	1	(9)
Matrices	10	(9)
Matrix Multiplication	19	(12)
Algebraic Properties of Matrix Operations	31	(8)
Special Types of Matrices and Partitioned Matrices	39	(11)
Matrix Transformations	50	(8)
Computer Graphics (Optional)	58	(8)
Correlation Coefficient (Optional)	66	(11)
Supplementary Exercises	73	(4)
Solving Linear Systems	77	(47)
Echelon Form of a Matrix	77	(24)
Elementary Matrices; Finding A-1	101	(9)
Equivalent Matrices	110	(4)
LU-Factorization (Optional)	114	(10)
Supplementary Exercises	121	(3)
Real Vector Spaces	124	(95)
Vectors in the Plane and in 3-Space	124	(11)
Vector Spaces	135	(7)
Subspaces	142	(12)
Span and Linear Independence	154	(10)
Basis and Dimension	164	(14)
Homogeneous Systems	178	(8)
Coordinates and Isomorphisms	186	(15)
Rank of a Matrix	201	(18)
Supplementary Exercises	214	(5)
Inner Product Spaces	219	(69)
Length and Direction in R2 and R3	219	(9)
Cross Product in R3 (Optional)	228	(7)
Inner Product Spaces	235	(13)
Gram-Schmidt Process	248	(12)
Orthogonal Complements	260	(16)
Least Squares (Optional)	276	(12)
Supplementary Exercises	283	(5)
Linear Transformations and Matrices	288	(70)
Definition and Examples	288	(11)
Kernel and Range of a Linear Transformation	299	(15)
Matrix of a Linear Transformation	314	(10)
Vector Space of Matrices and Vector Space of Linear Transformations (Optional)	324	(7)
Similarity	331	(7)
Introduction to Homogeneous Coordinates (Optional)	338	(20)
Supplementary Exercises	354	(4)
Determinants	358	(35)
Definition	358	(5)
Properties of Determinants	363	(10)
Cofactor Expansion	373	(7)
Inverse of a Matrix	380	(4)
Other Applications of Determinants	384	(6)
Determinants from a Computational Point of View	390	(3)
Supplementary Exercises	391	(2)
Eigenvalues and Eigenvectors	393	(101)
Eigenvalues and Eigenvectors	393	(17)
Diagonalization and Similar Matrices	410	(10)
Stable Age Distribution in a Population; Markov Processes (Optional)	420	(7)
Diagonalization of Symmetric Matrices	427	(12)
Spectral Decomposition and Singular Value Decomposition (Optional)	439	(11)
Real Quadratic Forms	450	(10)
Conic Sections	460	(8)
Quadric Surfaces	468	(9)
Dominant Eigenvalue and Principal Component Analysis (Optional)	477	(17)
Supplementary Exercises	491	(3)
Differential Equations (Optional)	494	(22)
Differential Equations	494	(10)
Dynamical Systems	504	(12)
Supplementary Exercises	513	(3)
Matlab for Linear Algebra	516	(35)
Input and Output in Matlab	517	(5)
Matrix Operations in Matlab	522	(3)
Matrix Powers and Some Special Matrices	525	(3)
Elementary Row Operations in Matlab	528	(10)
Matrix Inverses in Matlab	538	(1)
Vectors in Matlab	539	(2)
Applications of Linear Combinations in Matlab	541	(4)
Linear Transformations in Matlab	545	(2)
Matlab Command Summary	547	(4)
Matlab Exercises	551	(18)
Appendix A: Preliminaries	569	(4)
A.1 Sets	569	(1)
A.2 Functions	569	(4)
Appendix B: Complex Numbers	573	(17)
B.1 Complex Numbers	573	(8)
B.2 Complex Numbers in Linear Algebra	581	(9)
Appendix C: Introduction to Proofs	590	(9)
C.1 Logic	590	(5)
C.2 Techniques of Proof	595	(4)
Answers to Odd-Numbered Exercises	599
Index	1	(8)
Page Index to Lemmas, Theorems, and Corollaries	9
Photo Credits	1

Excerpts

Linear algebra continues to be an important course for a diverse number of students for at least two reasons. First, few subjects can claim to have such widespread applications in other areas of mathematics--multivariable calculus, differential equations, and probability, for example--as well as in physics, biology, chemistry, economics, finance, psychology, sociology, and all fields of engineering. Second, the subject presents the student at the sophomore level with an excellent opportunity to learn how to handle abstract concepts. This book provides an introduction to the basic ideas and computational techniques of linear algebra at the sophomore level. It includes carefully selected applications. The book introduces the student to working with abstract concepts: this includes an introduction to how to read and write proofs. In covering the basic ideas of linear algebra, the abstract ideas are carefully balanced by the considerable emphasis on the geometrical and computational aspects of the subject. This edition continues to provide the optional opportunity to use MATLAB or other software to enhance the practical side of linear algebra. What's New in the Eighth Edition We have been very pleased by the wide acceptance of the first seven editions of this book throughout the 34 years of its life. In preparing this edition, we have carefully considered many suggestions from faculty and students for improving the content and presentation of the material. Although a great many changes have been made to develop this major revision, our objective has remained the same as in the first seven editions:to present the basic ideas of linear algebra an a manner that the student will find understandable.To achieve this objective, the following features have been developed in this edition: Old Chapter 1,Linear Equations and Matrices,has been split into two chapters to improve pedagogy. Matrix multiplication is now covered more carefully in a separate section, Section 1.3. Section 1.6,Matrix Transformations,new to this edition, introduces at a very early stage some geometric applications. Section 1.7,Computer Graphics,has been moved from old Chapter 4 to give an application of matrix transformations. Several sections in old Chapters 1 and 4 have been moved to improve the organization, exposition, and flow of the material. Section 1.8,Correlation Coefficient,new to this edition, gives an application of the dot product to statistics. Section 5.6,Introduction to Homogeneous Coordinates,new to this edition, extends and generalizes earlier work on computer graphics. Section 7.9,Dominant Eigenvalue and Principal Component Analysis,news to this edition, includes several applications of this material. One of the applications discussed here is the way in which the highly successful search engine Google uses the dominant eigenvalue of an enormously large matrix to search the Web. Appendix C,Introduction to Proofs,new to this edition, provides a brief introduction to proofs in mathematics. The geometrical aspects of linear algebra have been greatly enhanced with 55 new figures added to this edition. More exercises at all levels have been added. Eigenvalues are now defined in terms of both real and complex numbers. MATLAB M-files have been upgraded to more modern versions. Key Terms have been added at the end of each section, reflecting the increased emphasis in mathematics on communication skills. A Chapter Review consisting of true/false questions and a quiz has been added to each chapter. EXERCISES The exercises form an integral part of the text. Many of them are numerical in nature, whereas others are of a theoretical type. The theoretical exercises (as well as many numerical ones) call for a verbal solution. In thi