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 fifth edition of Linear Algebra is 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. Throughout, we emphasize the symbiotic relationship between linear transformations and matrices. However, where appropriate, theorems are stated in the more general infinite-dimensional case. This enables us to apply the basic theory of vector spaces and linear transformations to find solutions to a homogeneous linear differential equation as well as the best approximation by a trigonometric polynomial to a continuous function.
Although the only formal prerequisite for this book is a one-year course in calculus, it requires the mathematical sophistication of typical junior and senior mathematics majors. This book is especially suited to 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 for use in a number of different courses (ranging from three to eight semester hours in length). 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 there 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 one-semester 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 one-semester four-hour course for students who have had some prior exposure to linear algebra.
Chapter 1 of the book presents the basic theory of vector spaces: sub- spaces, linear combinations, linear dependence and independence, bases, and dimension. The chapter concludes with an optional section in which we prove that every infinite-dimensional vector space has a basis.
Linear transformations and their relationships to matrices are the subjects of Chapter 2. We discuss the null space, range, and matrix representations of a linear transformation, isomorphisms, and change of coordinates. The chapter ends with optional sections on dual spaces and homogeneous linear differential equations.
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 once were. In a short course (less than one year), we prefer to treat determinants lightly so that more time may be devoted to the material in Chapters 5 through 7. Consequently, we have presented two alternatives in Chapter 4—a complete development of the theory (Sections 4.1 through 4.3) and a summary of important facts that are needed for the remaining chapters (Section 4.4). Optional Section 4.5 presents an axiomatic development of the determinant.
Chapter 5 discusses eigenvalues, eigenvectors, and diagonalization. One of the most important applications of this material occurs in computing matrix limits. We have therefore included an optional section on matrix limits and Markov chains in this chapter even though the most general statement of some of the results requires a knowledge of the Jordan canonical form. Section 5.4 contains material on invariant subspaces and the Cayley-Hamilton theorem.
Inner product spaces are the subject of Chapter 6. The basic mathematical theory (inner products; the Gram-Schmidt process; orthogonal complements; the adjoint of an operator; normal, self-adjoint, orthogonal and unitary operators; orthogonal projections; and the spectral theorem) is contained in Sections 6.1 through 6.6. Sections 6.7 through 6.11 contain diverse applications of the rich inner product space structure.
There are five appendices. The first four, which discuss sets, functions, fields, and complex numbers, respectively, are intended to review basic ideas used throughout the book. Appendix E on polynomials is used primarily in Chapters 5 and 7, especially in Section 7.4. We prefer to cite particular results from the appendices as needed rather than to discuss the appendices independently.
The following diagram illustrates the dependencies among the various chapters.
One final word is required about our notation. Sections and subsections labeled with an asterisk (*) are optional and may be omitted as the instructor sees fit. An exercise accompanied by the dagger symbol (†) is not optional, however—we use this symbol to identify an exercise that is cited in some later section that is not optional.
The organization of the fifth edition is essentially the same as its predecessor. Nevertheless, this edition contains many significant local changes that improve the book. Several of these streamline the presentation, and others clarify expositions that had led to student misunderstandings. Further improvements include revised proofs of some theorems, additional examples, new exercises, and literally hundreds of minor editorial changes. Many of these changes were prompted by George Bergman (University of California, Berkeley), who provided detailed comments about his experiences teaching from earlier editions, as well as numerous other professors and students who have written us with questions and comments about the fourth edition.
As an additional aid to students, we have made available online a solution to one theoretical exercise in each section of the book. These exercises each have their exercise number printed within a gray colored box, and the last sentence of each of these exercises gives a short URL for its online solution. See, for example, Exercise 5 on page 6.
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