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Summary
This topselling, theoremproof book presents a careful treatment of the principle topics of linear algebra, and illustrates the power of the subject through a variety of applications. It emphasizes the symbiotic relationship between linear transformations and matrices, but states theorems in the more general infinitedimensional case where appropriate.Chapter topics cover vector spaces, linear transformations and matrices, elementary matrix operations and systems of linear equations, determinants, diagonalization, inner product spaces, and canonical forms.For statisticians and engineers.
Table of Contents
Preface 

ix  


1  (63) 


1  (5) 


6  (10) 


16  (8) 

Linear Combinations and Systems of Linear Equations 


24  (11) 

Linear Dependence and Linear Independence 


35  (7) 


42  (16) 

Maximal Linearly Independent Subsets 


58  (6) 


62  (2) 

Linear Transformations and Matrices 


64  (83) 

Linear Transformations, Null Spaces, and Ranges 


64  (15) 

The Matrix Representation of a Linear Transformation 


79  (7) 

Composition of Linear Transformations and Matrix Multiplication 


86  (13) 

Invertibility and Isomorphisms 


99  (11) 

The Change of Coordinate Matrix 


110  (9) 


119  (8) 

Homogeneous Linear Differential Equations with Constant Coefficients 


127  (20) 


145  (2) 

Elementary Matrix Operations and Systems of Linear Equations 


147  (52) 

Elementary Matrix Operations and Elementary Matrices 


147  (5) 

The Rank of a Matrix and Matrix Inverses 


152  (16) 

Systems of Linear EquationsTheoretical Aspects 


168  (14) 

Systems of Linear EquationsComputational Aspects 


182  (17) 


198  (1) 


199  (46) 


199  (10) 


209  (13) 

Properties of Determinants 


222  (10) 

SummaryImportant Facts about Determinants 


232  (6) 

A Characterization of the Determinant 


238  (7) 


244  (1) 


245  (84) 

Eigenvalues and Eigenvectors 


245  (16) 


261  (22) 

Matrix Limits and Markov Chains 


283  (30) 

Invariant Subspaces and the CayleyHamilton Theorem 


313  (16) 


328  (1) 


329  (153) 


329  (12) 

The GramSchmidt Orthogonalization Process and Orthogonal Complements 


341  (16) 

The Adjoint of a Linear Operator 


357  (12) 

Normal and SelfAdjoint Operators 


369  (10) 

Unitary and Orthogonal Operators and Their Matrices 


379  (19) 

Orthogonal Projections and the Spectral Theorem 


398  (7) 

The Singular Value Decomposition and the Pseudoinverse 


405  (17) 

Bilinear and Quadratic Forms 


422  (29) 

Einstein's Special Theory of Relativity 


451  (13) 

Conditioning and the Rayleigh Quotient 


464  (8) 

The Geometry of Orthogonal Operators 


472  (10) 


480  (2) 


482  (66) 

The Jordan Canonical Form I 


482  (15) 

The Jordan Canonical Form II 


497  (19) 


516  (8) 

The Rational Canonical Form 


524  (24) 
Index of Definitions 

548  (1) 
Appendices 

549  (22) 


549  (2) 


551  (1) 


552  (3) 


555  (6) 


561  (10) 
Answers to Selected Exercises 

571  (18) 
Index 

589  
Excerpts
The language and concepts of matrix theory and, more generally, of linear algebra have come into widespread usage in the social and natural sciences, computer science, and statistics. In addition, linear algebra continues to be of great importance in modern treatments of geometry and analysis. The primary purpose of this fourth edition ofLinear Algebrais to present a careful treatment of the principal topics of linear algebra and to illustrate the power of the subject through a variety of applications. Our major thrust emphasizes the symbiotic relationship between linear transformations and matrices. However, where appropriate, theorems are stated in the more general infinitedimensional case. For example, this theory is applied to finding solutions to a homogeneous linear differential equation and the best approximation by a trigonometric polynomial to a continuous function. Although the only formal prerequisite for this book is a oneyear course in calculus, it requires the mathematical sophistication of typical junior and senior mathematics majors. This book is especially suited for a second course in linear algebra that emphasizes abstract vector spaces, although it can be used in a first course with a strong theoretical emphasis. The book is organized to permit a number of different courses (ranging from three to eight semester hours in length) to be taught from it. The core material (vector spaces, linear transformations and matrices, systems of linear equations, determinants, diagonalization, and inner product spaces) is found in Chapters 1 through 5 and Sections 6.1 through 6.5. Chapters 6 and 7, on inner product spaces and canonical forms, are completely independent and may be studied in either order. In addition, throughout the book are applications to such areas as differential equations, economics, geometry, and physics. These applications are not central to the mathematical development, however, and may be excluded at the discretion of the instructor. We have attempted to make it possible for many of the important topics of linear algebra to be covered in a onesemester course. This goal has led us to develop the major topics with fewer preliminaries than in a traditional approach. (Our treatment of the Jordan canonical form, for instance, does not require any theory of polynomials.) The resulting economy permits us to cover the core material of the book (omitting many of the optional sections and a detailed discussion of determinants) in a onesemester fourhour course for students who have had some prior exposure to linear algebra. Chapter 1 of the book presents the basic theory of vector spaces: subspaces, linear combinations, linear dependence and independence, bases, and dimension. The chapter concludes with an optional section in which eve prove that every infinitedimensional vector space has a basis. Linear transformations and their relationship to matrices are the subject of Chapter 2. We discuss the null space and range of a linear transformation, matrix representations of a linear transformation, isomorphisms, and change of coordinates. Optional sections on dual spaces and homogeneous linear differential equations end the chapter. The application of vector space theory and linear transformations to systems of linear equations is found in Chapter 3. We have chosen to defer this important subject so that it can be presented as a consequence of the preceding material. This approach allows the familiar topic of linear systems to illuminate the abstract theory and permits us to avoid messy matrix computations in the presentation of Chapters 1 and 2. There are occasional examples in these chapters, however, where we solve systems of linear equations. (Of course, these examples are not a part of the theoretical development.) The necessary background is contained in Section 1.4. Determinants, the subject of Chapter 4, are of much less importance than they