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Mathematics

 

Review Essays of Academic, Professional & Technical Books in the Humanities & Sciences

 

College Algebra Demystified by Rhonda Huettenmueller (McGraw-Hill Professional) One of the most valuable tools acquired in a university education, college algebra is essential for courses from the sciences to computing, engineering to mathematics. It can help you do better on placement exams, even before college, and it's useful in solving the computations of daily life. Now anyone with an interest in college algebra can master it. In College Algebra Demystified, entertaining author and experienced teacher Rhonda Huettenmueller breaks college algebra down into manageable bites with practical examples, real data, and a new approach that banishes algebra's mystery.

With College Algebra Demystified, you master the subject

one simple step at a time — at your own speed. Unlike most books on college algebra, general concepts are presented first — and the details follow. In order to make the process as clear and simple as possible, long computations are presented in a logical, layered progression with just one execution per step.

This fast and entertaining self-teaching course will help you —

  • Perform better on placement exams

  • Avoid confusion with detailed examples and solutions that help you every step of the way

  • Conquer the coordinate plane, lines and intercepts, parabolas' and nonlinear equations

  • Get comfortable with functions, graphs of functions, logarithms, exponents, and more

  • Master aspects of algebra that will help you with calculus, geometry,

  • trigonometry, physics, chemistry, computing, and engineering

  • Reinforce learning and pinpoint weaknesses with questions at the end of every chapter, and a final at the end of the book

Beginning Algebra (4th Edition) by K. Elayn Martin-Gay (Prentice Hall) Beginning Algebra, Fourth Edition was written to provide a solid foundation in algebra for students who might have had no previous experience in algebra. Specific care has been taken to ensure that students have the most up-to-date and relevant text preparation for their next mathematics course, as well as to help stu­dents to succeed in nonmathematical courses that require a grasp of algebraic fun­damentals. I have tried to achieve this by writing a user-friendly text that is keyed to objectives and contains many worked-out examples. The basic concepts of graph­ing are introduced early, and problem solving techniques, real-life and real-data applications, data interpretation, appropriate use of technology, mental mathemat­ics, number sense, critical thinking, decision-making, and geometric concepts are emphasized and integrated throughout the book.

The new edition includes an increased emphasis on study and test preparation skills. In addition, the fourth edition now includes a new resource, the Chapter Test Prep Video CD. With this CD/Video, students have instant access to video solutions for each of the chapter test questions contained in the text. It is designed to help them study efficiently.

The many factors that contributed to the success of the previous editions have been retained. In preparing this edition, I considered the comments and suggestions of colleagues throughout the country, students, and many users of the prior editions. The AMATYC Crossroads in Mathematics: Standards for Introductory College Mathematics before Calculus and the MAA and NCTM standards (plus Addenda), together with advances in technology, also influenced the writing of this text.

Beginning Algebra, Fourth Edition is part of a series of texts that can include Basic College Mathematics, Second Edition, Prealgebra, Fourth Edition, Intermediate Algebra, Fourth Edition, or Intermediate Algebra: A Graphing Approach, Third Edition, and Beginning and Intermediate Algebra, Third Edition, a combined alge­bra text. Throughout the series, pedagogical features are designed to develop stu­dent proficiency in algebra and problem solving, and to prepare students for future courses.

The following new features have been added to the fourth edition.

  • Increased Emphasis On Study Skills: New! Study Skills Reminders integrated throughout the text to help students hone their study skills and serve as a point-of-use support resource, reinforcing the skills covered in Section 1.1, Tips for Success in Mathematics.

  • New! Integrated Reviews serve as mid-chapter reviews and help students assimilate new skills and concepts they have learned separately over several sections. The reviews provide students with another opportunity to practice with mixed exercis­es as they master the topics.

  • Enhanced Section Exercise Sets: Mixed Practice Exercises. Exercise sets have been reorganized to include mixed practice exercises where appropriate. They give students the chance to assimilate the concepts and skills covered in separate objectives. Students have the opportu­nity to practice the kind of decision making they will encounter on tests.

  • New! Concept Extension Exercises. These have been added to the end of the section exercise sets. They extend the concepts and require students to combine sever-al skills or concepts. They expose students to the way math ideas build upon each other and offer the opportunity for additional challenges.

  • More Opportunities For Students To Check Their Understanding: New! Concept Checks are special exercises found in most sections following key examples. Working these will help students check their grasp of the concept being developed before moving to the next example.

  • Increased Emphasis On Improving Test Preparation: New! Chapter Test Prep Video CD packaged with each text and presented by Elayn Martin-Gay provides students with a resource to take and correct sample tests as they prepare for exams. Step-by-Step solutions are presented for every Chapter Test exercise contained in the text. Easy video navigation allows students to instantly access the solutions to the exact exercises they need help with.

The following key features have been retained from previous editions.

  • Readability and Connections I have tried to make the writing style as clear as pos­sible while still retaining the mathematical integrity of the content. When a new topic is presented, an effort has been made to relate the new ideas to those that students may already know. Constant reinforcement and connections within problem solving strategies, data interpretation, geometry, patterns, graphs, and situations from every-day life can help students gradually master both new and old information.

  • Problem Solving Process This is formally introduced in Chapter 2 with a four-step process that is integrated throughout the text. The four steps are Understand, Translate, Solve, and Interpret. The repeated use of these steps throughout the text in a variety of examples shows their wide applicability. Reinforcing the steps can increase students' confidence in tackling problems.

  • Applications and Connections Every effort was made to include as many acces­sible, interesting, and relevant real-life applications as possible throughout the text in both worked-out examples and exercise sets. The applications strengthen students' understanding of mathematics in the real world and help to motivate students. They show connections to a wide range of fields including agriculture, allied health, art, astronomy, automotive ownership, aviation, biology, business, chemistry, communication, computer technology, construction, consumer affairs, demographics, earth science, education, entertainment, environmental issues, finance and economics, food service, geography, government, history, hobbies, labor and career issues, life science, medicine, music, nutrition, physics, political science, population, recreation, sports, technology, transportation, travel, weather, and important related mathematical areas such as geometry and statistics. (See the Index of Applications on page xvi.) Many of the applications are based on recent and interesting real-life data. Sources for data include newspapers, magazines, government publications, publicly held companies, special interest groups, research organizations, and reference books. Opportunities for obtaining your own real data are also included.

  • Helpful Hints Helpful Hints contain practical advice on applying mathematical concepts. These are found throughout the text and strategically placed where stu­dents are most likely to need immediate reinforcement. They are highlighted in a box for quick reference and, as appropriate, an indicator line is used to precisely identify the particular part of a problem or concept being discussed. For instance, see pages 25 and 26.

  • Visual Reinforcement of Concepts The text contains numerous graphics, models, and illustrations to visually clarify and reinforce concepts. These include new and updated bar graphs, circle graphs in two and three dimensions, line graphs, calculator screens, application illustrations, photographs, and geometric figures.

  • Real World Chapter Openers The chapter openers focus on how math is used in a specific career, provide links to the World Wide Web, and reference a "Spotlight on Decision Making" feature within the chapter for further exploration of the career and the relevance of algebra.

  • Student Resource Icons At the beginning of each exercise set, videotape, tutorial software CD-Rom, Student Solutions Manual, Study Guide, and tutor center icons are displayed. These icons help reinforce that these learning aids are available should students wish to use them to review concepts and skills at their own pace. These items have direct correlation to the text and emphasize the text's methods of solution.

  • Chapter Highlights Found at the end of each chapter, the Chapter Highlights contain key definitions, concepts, and examples to help students understand and retain what they have learned.

  • Chapter Project This feature occurs at the end of each chapter, often serving as a chapter wrap-up. For individual or group completion, the multi-part Chapter Project, usually hands-on or data based, allows students to problem solve. 

Introductory Linear Algebra: An Application-Oriented First Course (8th Edition) by Bernard Kolman, David R. Hill (Prentice Hall) This book presents an introduction to linear algebra and to some of its significant applications. It is designed for a course at the freshman or sophomore level. There is more than enough material for a semester or quarter course. By omitting certain sections, it is possible in a one-semester or quarter course to cover the essentials of linear algebra (including eigenvalues and eigenvectors), to show how the computer is used, and to explore some applications of linear algebra. It is no exaggeration to say that with the many applications of linear algebra in other areas of mathematics, physics, biology, chemistry, engineering, statistics, economics, finance, psychology, and sociology, linear algebra is the undergraduate course that will have the most impact on students' lives. The level and pace of the course can be readily changed by varying the amount of time spent on the theoretical material and on the applications. Calculus is not a prerequisite; examples and exercises using very basic calculus are included and these are labeled "Calculus Required."

The emphasis is on the computational and geometrical aspects of the subject, keeping abstraction to a minimum. Thus we sometimes omit proofs of difficult or less-rewarding theorems while amply illustrating them with examples. The proofs that are included are presented at a level appropriate for the student. We have also devoted our attention to the essential areas of linear algebra; the book does not attempt to cover the subject exhaustively.

What Is New in the Eighth Edition

We have been very pleased by the widespread acceptance of the first seven editions of this book. The reform movement in linear algebra has resulted in a number of techniques for improving the teaching of linear algebra. The Linear Algebra Curriculum Study Group and others have made a number of important recommendations for doing this. In preparing the present edition, we have considered these recommendations as well as suggestions from faculty and students. Although many changes have been made in this edition, our objective has remained the same as in the earlier editions:

  1. to develop a textbook that will help the instructor to teach and the student to learn the basic ideas of linear algebra and to see some of its applications.

  2. To achieve this objective, the following features have been developed in this edition:

New sections have been added as follows:

  • Section 1.5, Matrix Transformations, introduces at a very early stage some geometric applications.

  • Section 2.1, An Introduction to Coding, along with supporting material on bit matrices throughout the first six chapters, provides an introduction to the basic ideas of coding theory.

  • Section 7.3, More on Coding, develops some simple codes and their basic properties related to linear algebra.

  • More geometric material has been added.

  • New exercises at all levels have been added. Some of these are more open-ended, allowing for exploration and discovery, as well as writing. More illustrations have been added.

  • MATLAB M-files have been upgraded to more modem versions.

  • Key terms have been added at the end of each section, reflecting the in-creased emphasis in mathematics on communication skills.

  • True/false questions now ask the student to justify his or her answer, providing an additional opportunity for exploration and writing.

  • Another 25 true/false questions have been added to the cumulative review at the end of the first ten chapters.

  • A glossary, new to this edition, has been added.

Exercises

The exercises in this book are grouped into three classes. The first class, Exercises, contains routine exercises. The second class, Theoretical Exercises, includes exercises that fill in gaps in some of the proofs and amplify material in the text. Some of these call for a verbal solution. In this technological age, it is especially important to be able to write with care and precision; therefore, exercises of this type should help to sharpen such skills. These exercises can also be used to raise the level of the course and to challenge the more capable and interested student. The third class consists of exercises developed by David R. Hill and are labeled by the prefix ML (for MATLAB). These exercises are designed to be solved by an appropriate computer software package.

Answers to all odd-numbered numerical and ML exercises appear in the back of the book. At the end of Chapter 10, there is a cumulative review of the introductory linear algebra material presented thus far, consisting of 100 true/false questions (with answers in the back of the book). The Instructor's Solutions Manual, containing answers to all even-numbered exercises and solutions to all theoretical exercises, is available (to instructors only) at no cost from the publisher. 

Presentation

We have learned from experience that at the sophomore level, abstract ideas must be introduced quite gradually and must be supported by firm foundations. Thus we begin the study of linear algebra with the treatment of matrices as mere arrays of numbers that arise naturally in the solution of systems of linear equations—a problem already familiar to the student. Much attention has been devoted from one edition to the next to refine and improve the pedagogical aspects of the exposition. The abstract ideas are carefully balanced by the considerable emphasis on the geometrical and computational foundations of the subject.

Material Covered

Chapter 1 deals with matrices and their properties. Section 1.5, Matrix Trans-formations, new to this edition, provides an early introduction to this important topic. This chapter is comprised of two parts: The first part deals with matrices and linear systems and the second part with solutions of linear systems. Chapter 2 (optional) discusses applications of linear equations and matrices to the areas of coding theory, computer graphics, graph theory, electrical circuits, Markov chains, linear economic models, and wavelets. Section 2.1, An Introduction to Coding, new to this edition, develops foundations for introducing some basic material in coding theory. To keep this material at a very elementary level, it is necessary to use lengthier technical discussions. Chapter 3 presents the basic properties of determinants rather quickly. Chapter 4 deals with vectors in R". In this chapter we also discuss vectors in the plane and give an introduction to linear transformations. Chapter 5 (optional) provides an opportunity to explore some of the many geometric ideas dealing with vectors in R2 and R3; we limit our attention to the areas of cross product in R3 and lines and planes.

In Chapter 6 we come to a more abstract notion, that of a vector space. The abstraction in this chapter is more easily handled after the material covered on vectors in R". Chapter 7 (optional) presents three applications of real vector spaces: QR-factorization, least squares, and Section 7.3, More on Cod-

Dew to this edition, introducing some simple codes. Chapter 8, on eigen-values and eigenvectors, the pinnacle of the course, is now presented in three sections to improve pedagogy. The diagonalization of symmetric matrices is carefully developed.

Chapter 9 (optional) deals with a number of diverse applications of eigen-values and eigenvectors. These Include the Fibonacci sequence, differential equations, dynamical systems, quadratic forms, conic sections, and quadric surfaces. Chapter 10 covers linear transformations and matrices. Section 10.4 (optional), Introduction to Fractals, deals with an application of a certain non-linear transformation. Chapter 11 (optional) discusses linear programming, an important application of linear algebra. Section 11.4 presents the basic ideas of the theory of games. Chapter 12, provides a brief introduction to MATLAB (which stands for MATRIX LABORATORY), a very useful software package for linear algebra computation, described below.

Appendix A covers complex numbers and introduces, in a brief but thorough manner, complex numbers and their use in linear algebra. Appendix B presents two more advanced topics in linear algebra: inner product spaces and composite and invertible linear transformations.

Applications

Most of the applications are entirely independent; they can be covered either after completing the entire introductory linear algebra material in the course or they can be taken up as soon as the material required for a particular application has been developed. Brief Previews of most applications are given at appropriate places in the book to indicate how to provide an immediate application of the material just studied. The chart at the end of this Preface, giving the prerequisites for each of the applications, and the Brief Previews will be helpful in deciding which applications to cover and when to cover them.

Some of the sections in Chapters 2, 5, 7, 9, and 11 can also be used as in-dependent student projects. Classroom experience with the latter approach has met with favorable student reaction. Thus the instructor can be quite selective both in the choice of material and in the method of study of these applications.

End of Chapter Material

Every chapter contains a summary of Key Ideas for Review, a set of supplementary exercises (answers to all odd-numbered numerical exercises appear in the back of the book), and a chapter test (all answers appear in the back of the book).

MATLAB Software

Although the ML exercises can be solved using a number of software pack-ages, in our judgment MATLAB is the most suitable package for this purpose. MATLAB is a versatile and powerful software package whose cornerstone is its linear algebra capability. MATLAB incorporates profession-ally developed quality computer routines for linear algebra computation. The code employed by MATLAB is written in the C language and is upgraded as new versions of MATLAB are released. MATLAB is available from The Math Works, Inc., 24 Prime Park Way, Natick, MA 01760, (508) 653-1415; e-mail: info@mathworks . corn and is not distributed with this book or the instructional routines developed for solving the ML exercises. The Student Edition of MATLAB also includes a version of Maple, thereby providing a symbolic computational capability.

Chapter 12 of this edition consists of a brief introduction to MATLAB's capabilities for solving linear algebra problems. Although programs can be written within MATLAB to implement many mathematical algorithms, it should be noted that the reader of this book is not asked to write programs. The user is merely asked to use MATLAB (or any other comparable soft-ware package) to solve specific numerical problems. Approximately 24 instructional M-files have been developed to be used with the ML exercises in this book and are available from the following Prentice Hall Web site: www. preanall.com/kolman. These M-files are designed to transform many of MATLAB's capabilities into courseware. This is done by providing pedagogy that allows the student to interact with MATLAB, thereby letting the student think through all the steps in the solution of a problem and relegating MATLAB to act as a powerful calculator to relieve the drudgery of a tedious computation. Indeed, this is the ideal role for MATLAB (or any other similar package) in a beginning linear algebra course, for in this course, more than in many others, the tedium of lengthy computations makes it almost impossible to solve a modest-size problem. Thus, by introducing pedagogy and reining in the power of MATLAB, these M-files provide a working partnership between the student and the computer. Moreover, the introduction to a powerful tool such as MATLAB early in the student's college career opens the way for other software support in higher-level courses, especially in science and engineering.

Algebraic Theory of Automata & Languages by Masami Ito (World Scientific Publishing Company) Although there are some books dealing with algebraic theory of automata, their contents consist mainly of Krohn–Rhodes theory and related topics. The topics in the present book are rather different. For example, automorphism groups of automata and the partially ordered sets of automata are systematically discussed. Moreover, some operations on languages and special classes of regular languages associated with deterministic and nondeterministic directable automata are dealt with. The book is self-contained and hence does not require any knowledge of automata and formal languages.

The theory of formal languages began with the classification of lan­guages by N. Chomsky in Syntactic Structures in 1957. Now, this classification is called the Chomsky hierarchy of languages. On the other hand, the theory of automata was initiated by M.O. Rabin and D. Scott in 1959. Their work can be regarded as the most important first step in the theory of automata in spite of its simplicity. Since then, these two fields have been developed by many researchers as the two most important theoretical foundations of computer sci­ence.

In this book, the author mainly handles formal languages and au­tomata from the algebraic point of view. In the first two chapters, Ito investigates the algebraic structure of automata and then he deals with a kind of global theory, that is, partially ordered sets of automata. In the following four chapters, he studies grammars, languages and operations on languages. In the last section, ito introduces special kinds of automata, i.e. directable automata. The subjects in the book seem to be unique compared to other books with similar titles. The contents of the book are based on the author's work which started in the mid 1970s. His work offers some alternative ways to conceptualize ideas in Wolfram’s New Kind of Science.

This book consists of 9 chapters:

  • In Chapter 1, Ito mainly deals with the automorphism groups of strongly connected automata and (n, G)-automata, that is represen­tations of strongly connected automata.

  • In Chapter 2, he generalizes the results in Chapter 1 to the class of general automata.

  • In Chapter 3, he considers partially ordered sets of automata where partial orders are induced by homomorphisms of automata.

  • In Chapter 4, he deals with the compositions and decompositions of regular languages under n-insertion and shuffle operations. Moreover, Ito considers a decidability problem with respect to the shuffle closures of regular commutative languages.

  • In Chapter 5, he determines the structure of a shuffle closed language.

  • In Chapter 6, insertion and deletion operations is treated in details.

  • In Chapter 7, shuffle and scattered deletion operations is dealt with.

  • In Chapter 8, first Ito provides the concept of directable au­tomata and then the deals with nondeterministic directable automata.

The Basics of Abstract Algebra by Paul E. Bland (W.H. Freeman) The text contains 12 chapters. The material in Chapter 0 is presented to establish notation and to provide a basis for review. The material on binary operations and congruence relations contained in this chapter may be new to some students, so these concepts should not be overlooked.

Chapter 1 is a study of the basic properties of the integers. The set of positive integers is used to define an order relation on the set of integers, and this construction is used as a model later in the text, when ordered integral domains are developed. Mathematical induction, introduced m this chapter, s hound be part of every student's mathematical toolbag. Often a student's understanding of induction is "prove a proposition true for I, assume it's true for k and prove it's true for k + 1." Such students think that this somehow proves the proposition holds for all positive integers. An explanation of how this particular version of induction can be arrived at from a complete statement of the Principle of Mathematical Induction is given in this chapter. The purpose is to give the student a better "picture" of induction. Chapter 1 provides an excellent starting point for students to begin the transition from computational mathematics to proof. Proving statements about properties of the integers provides students with concepts that can be easily understood and, at the same time, presents problems that require at least the beginnings of rigorous proof.

Chapters 2 and 3 introduce the student to the basic structures of abstract algebra: groups, rings, integral domains, division rings, and fields. In these chapters care has been taken to motivate the definitions with examples. These chapters contain basic material on groups, subgroups, normal subgroups, factor groups, group homomorphisms, rings, subrings, ideals, factor rings, and ring

homomorphisms. The concept of isomorphism is introduced before homomorphism for pedagogical reasons. Students seem to better understand one‑to‑one correspondence that preserves the binary operation(s) more quickly than homomorphism in general. This is particularly true when the isomorphic structures are finite and of small order.

Chapter 4 deals with the structure of our number systems. It shows how the field of rational numbers can be developed from the integral domain of integers, how the field of real numbers can be developed from the rational numbers, and how the field of complex numbers can be constructed from the field of real numbers. Including the construction of the real numbers system from the rational numbers in an abstract algebra book has proven to be somewhat controversial. Some argue that this topic does not belong in a book on abstract algebra but more properly belongs in a book on real analysis. However, for the sake of completeness, I have decided to cross this artificial boundary and include it here.

Chapters 5 and 6 present standard topics in abstract algebra. Chapter 5 returns to the study of groups with an investigation of permutation groups, cyclic groups, direct products of groups, and finite abelian groups. Chapter 6 begins with the development of polynomial rings via sequences of ring elements. The Factor Theorem and the Remainder Theorem are discussed in the second section, as are the greatest common divisor and the Euclidean algorithm for polynomials. The last section of this chapter presents a discussion of the Fundamental Theorem of Algebra and the Rational Root Theorem.

Chapter 7 covers standard topics on modular arithmetic in polynomial rings. When F is a field, the similarities between modular arithmetic in the polynomial ring F[xl and modular arithmetic in the ring Z of integers are pointed out. Euclidean Domains, Principal Ideal Domains, and Unique Factorization Domains are discussed, and the standard relationships among these algebraic structures are developed. Domains of quadratic integers are also investigated in Chapter 7.

Chapter 8 is an introduction to field extensions. Simple field extensions, algebraic field extensions, and the splitting field of a polynomial (whose coefficients lie in a field) are studied. Unfortunately every topic that one might wish to include in an undergraduate text in abstract algebra cannot be fully developed. This is the case with the algebraic closure of a field. The standard development of the algebraic closure of a field requires the use of concepts that are beyond the scope of this text. For this reason, a complete development of the algebraic clo­sure of a field is not provided. I believe, however, that because the field of com­plex numbers is the algebraic closure of the field of real numbers, a discussion of algebraic closure should be included. A discussion of the three famous problems of antiquity are also included in this chapter. It is shown that it is impossible to double a cube, to square a circle, and to trisect an angle using only a straightedge and compass. The inclusion of the material on these famous problems gives an interesting application of the theory of algebraic field extensions.

Chapter 9 is an introduction to Galois theory. The material is presented in a historical context in the sense that historical techniques are developed for find­ing the roots of polynomials of degree 2, 3, and 4. These are followed by a brief introduction to solvable groups. The remainder of the chapter concentrates on showing that, in general, it is not possible to solve polynomials of degree 5 or greater by radicals. The Fundamental Theorem of Galois Theory is presented for fields of characteristic 0 after first being illustrated by several examples. Chapter 9 concludes with a demonstration of the theorem via an example that establishes the Galois correspondence between the set of all subgroups of the Galois group and the intermediate fields of the splitting field of a polymonial.

Chapters 10 and 11 form an introduction to vector spaces and modules and give a standard treatment of these topics. Subspaces, submodules, factor mod­ules, and linear mappings are developed. Inner product spaces are also defined and investigated. Vector spaces are presented first in Chapter 10. Although every vector space is a module, studying these concepts separately demonstrates their similarities and differences more clearly Every vector space has a basis, but this is not true in general for modules. Examples are given of mod­ules that do not have bases. Moreover, the number of elements in a basis of a vector space is unique‑another property that fails to hold for all modules. Chapter 11 concludes with an interesting example of a module that has a basis with one element and a basis with two elements.

 Linear Algebra: A Geometric Approach by Ted Shifrin, Malcolm Ritchie Adams (W.H. Freeman) We begin Chapter 1 with a treatment of vectors, first in 1<82 and then in higher dimensions, emphasizing the interplay between algebra and geometry. Parametric equations of lines and planes and the notion of linear combination are introduced in the first section, and dot products in the second. We then treat systems of linear equations, starting with a discussion of hyperplanes in 18", and then introducing matrices and Gaussian elimination to arrive at reduced echelon form and the parametric representation of the general solution. We then

discuss consistency and the relation between solutions of the homogeneous and inhomogeneous systems. We conclude with a selection of applications.

In Chapter 2 we treat the mechanics of matrix algebra, including a first brush with 2 X 2 matrices as geometrically defined linear transformations. Multiplication of matrices is viewed as a generalization of multiplication of matrices by vectors, introduced in Chapter 1. We bring in elementary matrices and inverses, mention the LU decomposition, and introduce the notion of transpose.

The heart of the traditional linear algebra course enters in Chapter 3, where we deal with subspaces, linear independence, bases, and dimension. Orthogonality is a major theme throughout our discussion, as is the importance of the process of going back and forth between the parametric representation of a subspace of R" and its definition as the solution set of a homogeneous system of linear equations. In the fourth section, we officially give the algorithms for constructing bases for the four fundamental subspaces associated with a matrix. In the optional fifth section, we give the interpretation of these fundamental subspaces in the context of graph theory. In the sixth section, we discuss various examples of "abstract" vector spaces, concentrating on matrices, polynomials, and function spaces. The Lagrange interpolation formula is derived by defining an appropriate inner product on the vector space of polynomials.

In Chapter 4 we continue with the geometric flavor of the course, by discussing projections, least squares solutions of inconsistent systems, and orthogonal bases and the Gram‑Schmidt process. Motivated by projections and other geometrically defined mappings, we study linear transformations and the change‑of‑basis formula. Here we adopt the viewpoint that the matrix of a geometrically defined transformation is often easy to calculate in a coordinate system adapted to the geometry of the situation; then we can calculate its standard matrix by changing coordinates. The diagonalization problem emerges as natural, and we return to it fully in Chapter 6.

We give a more thorough treatment of determinants in Chapter 5 than is typical for introductory texts. We motivate the determinant by studying signed area in X82, characterize it by its behavior under row operations, and suggest that in X83 this is consistent with a calculation of signed volume. In the last section, we give the formula for expanding a determinant in cofactors and conclude with Cramer's Rule.

Chapter 6 is devoted to a thorough treatment of eigen-values, eigen-vectors, diagonalizability, and various applications. In the first section we introduce the characteristic polynomial, and in the second we introduce the notions of algebraic and geometric multiplicity and give a sufficient criterion for a matrix with real eigen-values to be diagonalizable. In the third section we solve some difference equations, emphasizing how eigen-values and eigenvectors give a "normal‑mode" decomposition of the solution. We conclude the section with an optional discussion of Markov processes and stochastic matrices. In the last section, we prove the Spectral Theorem, which we believe‑even in this most basic setting‑is one of the important theorems all mathematics majors should know; we include a brief discussion of its application to conics and quadric surfaces.

Chapter 7 is composed of three independent special topics. In the first section we discuss the two obstructions that arose in Chapter 6 to diagonalizing a matrix: complex eigen-values and repeated eigen-values. Although Jordan canonical form does not ordinarily appear in introductory texts, it is conceptually important and widely used in the study of systems of differential equations and dynamical systems. In the second section we give a brief introduction to the subject of affine transformations and projective geometry, including discussions of the isometries (motions) of R2 and 183. We discuss the notion of perspective projection, which is how computer graphics programs draw images on the screen. An amusing theoretical consequence of this discussion is the fact that circles, ellipses, parabolas, and hyperbolas are all "projectively equivalent" (i.e., can all be seen by projecting any one on different viewing screens). The third and last section is perhaps the most standard, presenting the matrix exponential and applications to systems of constant‑coefficient ordinary differential equations. Once again, eigen-values and eigenvectors play a central role in "uncoupling" the system and giving rise, physically, to normal modes.

Relative Homological Algebra by Edgar E. Enochs, Overtoun M. G. Jenda (De Gruyter Expositions in Mathematics, 30: Walter de Guyter) Excerpt from preface: The subject of relative homological algebra was introduced by S. Eilenberg and J. C. Moore in their 1965 AMS Memoir `Foundations of Relative Homological Algebra'. We now have in hand more theorems guaranteeing the existence of precovers, covers, preenvelopes and envelopes. These are basic objects of the subject and are used to construct resolutions and then left and right derived functors. Also, several new useful ideas have come into play since the appearance of Eilenberg and Moore's work. Among others these include the various versions of what is now known as Wakamatsu lemma, the notions of special precovers and preenvelopes and the orthogonality of classes of objects of an abelian category with respect to the extension function. Hence it seems opportune to now give a systematic treatment of this subject along with the new developments and applications.

Relative Homological Algebra is aimed at graduate students. For that reason, we have attempted to make the book a reasonably self-contained treatment of the subject requiring only familiarity with basic notions in module and ring theory at the level of Basic Algebra I by Nathan Jacobson.

The first three chapters give the basic tools and notation that will be used throughout the book. This material constitutes notes from our lectures at our respective universities and is suitable for an introductory course in module and ring theory.

The material in chapter four that deals with torsion free covers over integral domains is not essential to what follows in the book, but the ideas and proofs in this chapter give the flavor of what is to come. Chapter five gives information about precovers and covers and chapter six deals with preenvelopes and envelopes. Chapter seven introduces the notion of cotorsion theory that is used to prove the existence of special covers and envelopes. Chapter eight introduces balance (on the left or the right) of a function of two variables. Balance means that we have two specific kinds of resolutions of each of the two variables each of which can be used to compute the relative derived functions. We show that the basic functions Hom and Tensor are balanced using resolutions different from the usual projective, injective and flat resolutions. This allows us to compute useful versions of the Extension and Torsion functions with negative indices. We consider chapters five, six, seven and eight as the heart of the book. This material together with chapters four and nine is suitable for a course in relative homological algebra and its applications to commutative and noncommutative algebra.

The remainder of the book gives applications to ring theory and is more specialized. The commutative rings that we consider include local Cohen‑Macaulay rings admitting a dualizing module with the Gorenstein local rings as a special case. For example, we prove Auslander's announced (but unpublished) result concerning the existence of maximal Cohen‑Macaulay approximations over Gorenstein local rings. We also consider a noncommutative version of Gorenstein rings which we call Iwanaga-Gorenstein rings. Over these rings there is an especially pleasant application of relative homological algebra. We define relative versions of projective, injective and flat modules which we label Gorenstein. We prove that over an Iwanaga-Gorenstein ring there are enough Gorenstein projectives, injectives and flats (that is, the appropriate precovers and preenvelopes exist). We then show that Hom and Tensor are balanced when we use these Gorenstein versions of the projective, injective and flat modules to compute the resolutions over Iwanaga­Gorenstein rings. Then we prove that these rings have finite global dimension in this situation.

Algebra and Trigonometry, Second Edition by Robert Blitzer (Prentice Hall) was written Algebra and Trigonometry, Second Edition do help diverse students, with different backgrounds ant future goals do succeed. The book has three fundamental goals:

  1. To help students acquire a solid foundation in algebra ant trigonometry, preparing them for other courses such as calculus, business calculus, ant finite mathematics.

  2. To show students how algebra ant trigonometry can motel and solve authentic real-world problems.

  3. To enable students do develop problem-solving skills, while fostering critical thinking, within an interesting setting.

One major obstacle in the way of achieving these goals is the fact that very few students actually read their textbook. This has been a regular source of frustration for me ant my colleagues in the classroom. Anecdotal evidence gathered over years highlights two basic reasons that students to nod take advantage of their textbook:

•           "I'll never use this information."

•           "I can't follow the explanations."

As a result, I've written every page of this book with the intent of eliminating these two objections. See the book's Walk through, beginning on page xiv for the ideas ant tools I've used do to so.

A Brief Note on Technology

Technology, ant specifically the use of a graphing utility, is covered thoroughly, although ids coverage by an instructor is optional. If you require the use of a graphing utility in the course, you will find support for this approach, particularly in the wide selection of clearly designated technology exercises in each exercise set. If you wish do minimize or eliminate the discussion or use of a graphing utility, the book is written do enable you do to so. Regardless of the role technology plays in your course, the technology boxes with TI-83 screens that appear throughout the book should allow your students do understand what graphing utilities can to, enabling them do visualize, verify, or explore what they have already graphed or manipulated by hand. The book's technology coverage is intended do reinforce, bud never replace, algebraic solutions.

General Changes to the Second Edition

New Applications and Updated Real-World Data. Many new, innovative applications, supported by data that extend as far up to the present as possible, appear throughout the book.

Expanded Exercise Sets. There are new problems in many of the exercise sets. Some of these problems provide instructors with the option of creating assignments that take practice and application exercises from a basic level to a more challenging level than in the previous edition. In order to update applications and provide users with an ongoing selection of novel applications, many application problems from the previous edition were replaced with new exercises.

New Section Openers and Enrichment Essays. The Second Edition contains a variety of new section openers and enrichment essays, ranging from the five all-time celebrity winners on Jeopardy! (Section 2.3 opening scenario) to a comparison between the probability of dying and the probability of winning Florida 's lottery (Section 11.7 essay).

Increased Study Tip Boxes. The book's study tip boxes offer suggestions for problem solving, point out common errors to avoid, and provide informal hints and suggestions. These invaluable hints, including suggestions for review in preparation for the section ahead, appear in greater abundance in the Second Edition.

Expanded Technology. An increase in the number of optional technology boxes in the Second Edition illustrates the many capabilities of graphing utilities that go beyond just graphing functions.

New Chapter Review Grids. The chapter summaries, presented as outlines in the previous edition, are now organized into two-column review charts. The left column summarizes the definitions and concepts for every section of the chapter. The right column refers students to examples (by example number and page number) that illustrate these key concepts.

Expanded Supplements Package. The Second Edition is supported by a wealth of supplements designed for added effectiveness and efficiency, many of these new to this edition. (New supplements include MathPak 5 tutorial software now with trigonometry content and a diagnostic component; PH GradeAssist-an automated homework/assessment creation, delivery, and grading system; Instructor Resource CD ROM - contains all supplements for instructors in one easy location; and more.) See page x for details, under "Supplements" or ask your Prentice Hall representative for information.

Specific Content and Organizational Changes to the Second Edition

Section P.5 (Factoring Polynomials) now contains a brief discussion on factoring algebraic expressions containing fractional and negative exponents. This skill is helpful to students going on to calculus.

The discussion of complex numbers was moved from Chapter P, the prerequisites chapter, to Chapter 1, Section 1.4. This change enables students to immediately apply their understanding of complex numbers to their work in solving quadratic equations (Section 1.5).

The discussion of graphs and graphing utilities was moved from Chapter P to Chapter 1, Section 1.1. This nicely sets the stage for using graphing to support the algebraic work on solving equations and inequalities developed in Chapter 1.

Section 2.1 (Lines and Slope) now contains a discussion on parallel and perpendicular lines. In the previous edition, this material appeared in a section that also discussed circles. Presenting parallel and perpendicular lines in the section on lines and slope results in a more complete and unified discussion.

Section 2.2 (Distance and Midpoint Formulas; Circles) is now primarily devoted to circles. Distance and midpoint formulas, presented in Chapter P in the previous edition, were moved to this section because students need to use the distance formula to develop the formula for a circle.

Section 2.3 (Basics of Functions) now contains the definition of the difference quotient with an illustrative example.

Section 2.4 (Graphs of Functions) contains a new presentation on relative maximum and relative minimum values of a function, a natural outgrowth of the section's material on increasing and decreasing functions. There is also a new discussion of a function's average rate of change, with a relationship to the difference quotient from the previous section. These new topics are extremely important to students going on to calculus. They provide all students with an increased understanding of functions' graphs and how those graphs are changing.

Section 2.5 (Transformations of Functions) is now devoted exclusively to transformations, a difficult topic for many students. Unlike the previous edition, the section does not contain material on combinations of functions.

Section 2.6 (Combinations of Functions; Composite Functions) now includes combinations of functions and composite functions in one section. The section now tells a more coherent story - how to create new functions from given functions. With the section's emphasis on composite functions, new discussions on determining domains for composite functions and writing functions as compositions have been added.

Section 2.7 (Inverse Functions) is now devoted exclusively to the topic of inverse functions. This should appeal to users who prefer to cover inverse functions in Chapter 4, after Section 4.1 (Exponential Functions) and before Section 4.2 (Logarithmic Functions).

Section 3.6 (Rational Functions and Their Graphs) now contains a general discussion on cost and average cost functions. This change makes it possible for students to model these functions from verbal conditions before exploring the behavior of their graphs.

Section 7.6 (Vectors) and Section 7.7 (The Dot Product) have been slightly reorganized. The discussion on writing a vector in terms of its magnitude and direction was moved from Section 7.7 to Section 7.6. This move allows Section 7.7 to be devoted, without interruption, to the dot product. It also allows for a more thorough presentation in Section 7.6 on finding resultant forces by expressing a vector in terms of its magnitude and direction.

Section 8.1 (Systems of Linear Equations in Two Variables) contains a new application involving cost functions, revenue functions, and break-even points. This topic is important to business majors and gives students further practice in developing functions that model verbal conditions.

Section 9.3 (Matrix Operations and Their Applications) now includes a brief discussion on solving matrix equations. This topic serves as a nice application of matrix addition, scalar multiplication, and their properties.

Section 10.3 (The Parabola) now uses the latus rectum as part of the graphing strategy.

Section 11.5 (The Binomial Theorem) now gives the formula for the (r + 1)st term, rather than the rth term, of the expansion of (a + b)". Many students find the formula for the (r + 1)st term easier to work with when finding a particular term in a binomial expansion.

 

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