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Linear and Abstract Algebra Books.
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Linear and Abstract Algebra Books.
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Most books that use MATLAB are aimed at readers who know how to program. This book is for people who have never programmed before.
As a result, the order of presentation is unusual. The book starts with scalar values and works up to vectors and matrices very gradually. This approach is good for beginning programmers, because it is hard to understand composite objects until you understand basic programming semantics. But there are problems:
The MATLAB documentation is written in terms of matrices, and so are the error messages. To mitigate this problem, the book explains the necessary vocabulary early and deciphers some of the messages that beginners find confusing.
Many of the examples in the first half of the book are non-standard MATLAB. I address this problem in the second half by translating the examples into a more idiomatic style.
The book puts a lot of emphasis on functions, in part because they are an important tool for controlling program complexity, and also because they are useful for working with MATLAB tools like fzero and ode45.
I assume that readers know calculus, differential equations, and physics, but not linear algebra. I explain the math as I go along, but the descriptions might not be enough for someone who hasn't seen the material before.
There are small exercises within each chapter, and a few larger exercises at the end of some chapters.
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Users want ``black box''Black box: A piece of software that can be used without knowledge of its inner workings; the user supplies the input, and the output is more or less guaranteed to be correct. software that they can use with complete confidence for general problem classes without having to understand the fine algorithmic details.
* Users want to be able to tuneTune: Adapt software for a specific application and computing environment in order to obtain better performance in that case only. data structures for a particular application, even if the software is not as reliable as that provided for general methods.
It turns out both are true, for different groups of users.
Traditionally, users have asked for and been provided with black box software in the form of mathematical libraries such as LAPACK , LINPACK , NAG , and IMSL . More recently, the high-performance community has discovered that they must write custom software for their problem. Their reasons include inadequate functionality of existing software libraries, data structures that are not natural or convenient for a particular problem, and overly general software that sacrifices too much performance when applied to a special case of interest.
Can we meet the needs of both groups of users? We believe we can. Accordingly, in this book, we introduce the use of templates Template: Description of an algorithm, abstracting away from implementational details. A template is a description of a general algorithm rather than the executable object code or the source code more commonly found in a conventional software library. Nevertheless, although templates are general descriptions of key algorithms, they offer whatever degree of customization the user may desire. For example, they can be configured for the specific data structure of a problem or for the specific computing system on which the problem is to run.
We focus on the use of iterative methods for solving large sparse systems of linear equations.Iterative method: An algorithm that produces a sequence of approximations to the solution of a linear system of equations; the length of the sequence is not given a priori by the size of the system. Usually, the longer one iterates, the closer one is able to get to the true solution. See: Direct method.Direct method: An algorithm that produces the solution to a system of linear equations in a number of operations that is determined a priori by the size of the system. In exact arithmetic, a direct method yields the true solution to the system. See: Iterative method.
Many methods exist for solving such problems. The trick is to find the most effective method for the problem at hand. Unfortunately, a method that works well for one problem type may not work as well for another. Indeed, it may not work at all.
Thus, besides providing templates, we suggest how to choose and implement an effective method, and how to specialize a method to specific matrix types. We restrict ourselves to iterative methods, which work by repeatedly improving an approximate solution until it is accurate enough. These methods access the coefficient matrix of the linear system only via the matrix-vector product (and perhaps ). Thus the user need only supply a subroutine for computing (and perhaps ) given , which permits full exploitation of the sparsity or other special structure of .
We believe that after reading this book, applications developers will be able to use templates to get their program running on a parallel machine quickly. Nonspecialists will know how to choose and implement an approach to solve a particular problem. Specialists will be able to assemble and modify their codes-without having to make the huge investment that has, up to now, been required to tune large-scale applications for each particular machine. Finally, we hope that all users will gain a better understanding of the algorithms employed. While education has not been one of the traditional goals of mathematical software, we believe that our approach will go a long way in providing such a valuable service.
Why Use Templates?
Templates offer three significant advantages. First, templates are general and reusable. Thus, they can simplify ports to diverse machines. This feature is important given the diversity of parallel architectures.
Second, templates exploit the expertise of two distinct groups. The expert numerical analyst creates a template reflecting in-depth knowledge of a specific numerical technique. The computational scientist then provides ``value-added'' capability to the general template description, customizing it for specific contexts or applications needs.
And third, templates are not language specific. Rather, they are displayed in an Algol-like structure, which is readily translatable into the target language such as FORTRAN (with the use of the Basic Linear Algebra Subprograms, or BLAS , whenever possible) and C. By using these familiar styles, we believe that the users will have trust in the algorithms. We also hope that users will gain a better understanding of numerical techniques and parallel programming.
For each template, we provide some or all of the following:
* a mathematical description of the flow of the iteration;
* discussion of convergence and stopping criteria;
* suggestions for applying a method to special matrix types (e.g., banded systems);
* advice for tuning (for example, which preconditioners are applicable and which are not);
* tips on parallel implementations; and
* hints as to when to use a method, and why
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Contents:
1. Foundations
2. Elements of Representation Theory
3. Peirce Decompositions and Jordan Matrix Algebras
4. Jordan Algebras with Minimum Conditions on Quadratic Ideals
5. Structure Theory for Finite-Dimensional Jordan Algebras
6. Generic Minimum Polynomials, Traces and Norms
7. Representation Theory for Separable Jordan Algebras
8. Connections with Lie Algebras
9. Exceptional Jordan Algebras
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This is a WWW textbook written by Evans M. Harrell II and James V. Herod, both of Georgia Tech. It is suitable for a first course on partial differential equations, Fourier series and special functions, and integral equations. Students are expected to have completed two years of calculus and an introduction to ordinary differential equations and vector spaces. For recommended 10-week and 15-week syllabuses, read the preface.
This text concentrates on mathematical concepts rather than on details of calculations, which are often done with software, such as Maple or Mathematica. It is not necessary to have experience with Maple or Mathematica in order to read this text, nor is it the goal of this text to teach software, but there are links in the text to Maple worksheets and Mathematica notebooks, which perform calculations and provide some supplementary instructive material. The supplementary material exists both in a "flat" form, which can be read with Netscape, and also in an active form, requiring mathematical software.
The book is targeted at engineering students who have had two years of calculus, introductory linear algebra, and introductory ordinary differential equations. It has been used at Georgia Tech for a basic introduction to partial differential equations, Fourier series and other orthogonal series, integral equations, and linear operator theory, and our students have given the Web format high marks.
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This set of notes has been compiled over a period of some 25 years. Some chapters were used in various forms and on many occasions between 1981 and 1990 by the author at Imperial College, University of London. The remaining chapters were written in Sydney. All 12 chapters have been in use at Macquarie University since 1997.
The material has been organized in such a way to create a single volume suitable for use in the algebra half of the units MATH135, MATH136, MATH132, MATH133 and MATH235 at Macquarie University. The following is the suggested order for the presentation of the material:
MATH135 and MATH132:
* Chapters 1, 2, 3 and 4.
MATH136 and MATH133:
* Chapters 5, 6 and 7.
MATH235:
* Chapters 8, 9, 10, 11 and 12.
To read the notes, click the chapters below for connection to the appropriate PDF files. You will need Adobe Acrobat Reader Version 4.0 or later.
The material is available free to all individuals, on the understanding that it is not to be used for financial gains, and may be downloaded and/or photocopied, with or without permission from the author. However, the documents may not be kept on any information storage and retrieval system without permission from the author, unless such system is not accessible to any individuals other than its owners.
Chapter 1: LINEAR EQUATIONS (last uploaded on 28 February 2006)
* Introduction
* Elementary Row Operations
* Row Echelon Form
* Reduced Row Echelon Form
* Solving a System of Linear Equations
* Homogeneous Systems
* Application to Network Flow
* Application to Electrical Networks
* Application to Economics
* Application to Chemistry
* Application to Mechanics
Chapter 2: MATRICES (last uploaded on 10 April 2006)
* Introduction
* Systems of Linear Equations
* Inversion of Matrices
* Application to Matrix Multiplication
* Finding Inverses by Elementary Row Operations
* Criteria for Invertibility
* Consequences of Invertibility
* Application to Economics
* Matrix Transformation on the Plane
* Application to Computer Graphics
* Complexity of a Non-Homogeneous System
* Matrix Factorization
* Application to Games of Strategy
Chapter 3: DETERMINANTS (last uploaded on 19 April 2006)
* Introduction
* Determinants for Squares Matrices of Higher Order
* Some Simple Observations
* Elementary Row Operations
* Further Properties of Determinants
* Application to Curves and Surfaces
* Some Useful Formulas
* Further Discussion
Chapter 4: VECTORS (last uploaded on 4 May 2006)
* Introduction
* Vectors in 2-Space
* Vectors in 3-Space
* Vector Products
* Scalar Triple Products
* Application to Geometry in 3-Space
* Application to Mechanics
Chapter 5: INTRODUCTION TO VECTOR SPACES (last uploaded on 24 February 2005)
* Real Vector Spaces
* Subspaces
* Linear Combination
* Linear Independence
* Basis and Dimension
Chapter 6: VECTOR SPACES ASSOCIATED WITH MATRICES (last uploaded on 24 February 2005)
* Introduction
* Row Spaces
* Column Spaces
* Rank of a Matrix
* Nullspaces
* Solution of Non-Homogeneous Systems
Chapter 7: EIGENVALUES AND EIGENVECTORS (last uploaded on 24 February 2005)
* Introduction
* The Diagonalization Problem
* Some Remarks
* An Application to Genetics
Chapter 8: LINEAR TRANSFORMATIONS (last uploaded on 10 April 2006)
* Euclidean Linear Transformations
* Linear Operators on the Plane
* Elementary Properties of Euclidean Linear Transformations
* General Linear Transformations
* Change of Basis
* Kernel and Range
* Inverse Linear Transformations
* Matrices of General Linear Transformations
* Change of Basis
* Eigenvalues and Eigenvectors
Chapter 9: REAL INNER PRODUCT SPACES (last uploaded on 28 February 2006)
* Euclidean Inner Products
* Real Inner Products
* Angles and Orthogonality
* Orthogonal and Orthonormal Bases
* Orthogonal Projections
Chapter 10: ORTHOGONAL MATRICES (last uploaded on 28 February 2006)
* Introduction
* Eigenvalues and Eigenvectors
* Orthonormal Diagonalization
Chapter 11: APPLICATIONS OF REAL INNER PRODUCT SPACES (last uploaded on 28 February 2006)
* Least Squares Approximation
* Quadratic Forms
* Real Fourier Series
Chapter 12: COMPLEX VECTOR SPACES (last uploaded on 28 February 2006)
* Complex Inner Products
* Unitary Matrices
* Unitary Diagonalization
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This course is concerned with the fundamental structures underlying linear spaces and their applications.
The processes discussed are applicable in most areas of mathematics. By studying these basic structures, we obtain an insight into the ways of dealing with a wide variety of problems and of the forms of solutions we should expect.
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This book is an introduction to linear algebra for pre-calculus students. It is a stand-alone unit in the sense that no prior knowledge of matrices is assumed. Students with experience in general mathematics, up to and including Algebra I, should be able to comprehend the material. However, most students have not had experience with the topics in the latter chapters, so the pace of the course should allow for the students to spend extra time with these chapters.
We begin with chapters that explain the matrix operations of addition, subtraction, scalar multiplication, and matrix multiplication. These topics are covered in most pre-calculus texts that are currently in use. This unit also allows the students to explore the notions of inverse, determinant, and consistent and inconsistent systems; these topics are covered in some pre-calculus text books. Our unit also provides the students with an introduction to Markov chains, curve fitting, eigenpairs, and some of the numerical challenges that are encountered when matrices are used to solve real-world problems. These latter topics are rarely addressed in pre-calculus texts. The unit was created from elementary principles with significant input from Rice University faculty and students. Various current texts, recommendations from the National Council of Teachers of Mathematics (NCTM), and the Texas Essential Knowledge and Skills (TEKS) were examined in order to determine which topics should and should not be included in this text.
Because many books teach procedures rather than concepts, the students do not receive enough information to expand beyond the examples in the book. For example, some books teach methods which apply only to the special case of 2 by 2 matrices when they address the notions of inverse and determinant. However, this text presents methods for finding inverses and determinants of square matrices of any size. Since the students learn the concepts and these general methods, their knowledge is not restricted by the examples in the book.
This entire unit was written so that pre-calculus teachers and students will have a text that clearly and accurately explains the introductory concepts of linear algebra. It explores the topics that are currently addressed in pre-calculus courses, but emphasizes concepts rather than than just procedures. This unit also provides students with many more real-world linear algebra topics to explore than are presented in current texts. It is hoped that this unit will not only help students understand linear algebra, but will also spark an interest in, and an appreciation for, the mathematical sciences.
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Here is a listing of all the material that is currently available online.
Systems of Equations and Matrices
Systems of Equations In this section we’ll introduce most of the basic topics that we’ll need in order to solve systems of equations including augmented matrices and row operations.
Solving Systems of Equations Here we will look at the Gaussian Elimination and Gauss-Jordan Method of solving systems of equations.
Matrices We will introduce many of the basic ideas and properties involved in the study of matrices.
Matrix Arithmetic & Operations In this section we’ll take a look at matrix addition, subtraction and multiplication. We’ll also take a quick look at the transpose and trace of a matrix.
Properties of Matrix Arithmetic We will take a more in depth look at many of the properties of matrix arithmetic and the transpose.
Inverse Matrices and Elementary Matrices Here we’ll define the inverse and take a look at some of its properties. We’ll also introduce the idea of Elementary Matrices.
Finding Inverse Matrices In this section we’ll develop a method for finding inverse matrices.
Special Matrices We will introduce Diagonal, Triangular and Symmetric matrices in this section.
LU-Decompositions In this section we’ll introduce the LU-Decomposition a way of “factoring” certain kinds of matrices.
Systems Revisited Here we will revisit solving systems of equations. We will take a look at how inverse matrices and LU-Decompositions can help with the solution process. We’ll also take a look at a couple of other ideas in the solution of systems of equations.
Determinants
The Determinant Function We will give the formal definition of the determinant in this section. We’ll also give formulas for computing determinants of and matrices.
Properties of Determinants Here we will take a look at quite a few properties of the determinant function. Included are formulas for determinants of triangular matrices.
The Method of Cofactors In this section we’ll take a look at the first of two methods form computing determinants of general matrices.
Using Row Reduction to Find Determinants Here we will take a look at the second method for computing determinants in general.
Cramer’s Rule We will take a look at yet another method for solving systems. This method will involve the use of determinants.
Euclidean n-space
Vectors In this section we’ll introduce vectors in 2-space and 3-space as well as some of the important ideas about them.
Dot Product & Cross Product Here we’ll look at the dot product and the cross product, two important products for vectors. We’ll also take a look at an application of the dot product.
Euclidean n-Space We’ll introduce the idea of Euclidean n-space in this section and extend many of the ideas of the previous two sections.
Linear Transformations In this section we’ll introduce the topic of linear transformations and look at many of their properties.
Examples of Linear Transformations We’ll take a look at quite a few examples of linear transformations in this section.
Vector Spaces
Vector Spaces In this section we’ll formally define vectors and vector spaces.
Subspaces Here we will be looking at vector spaces that live inside of other vector spaces.
Span The concept of the span of a set of vectors will be investigated in this section.
Linear Independence Here we will take a look at what it means for a set of vectors to be linearly independent or linearly dependent.
Basis and Dimension We’ll be looking at the idea of a set of basis vectors and the dimension of a vector space.
Change of Basis In this section we will see how to change the set of basis vectors for a vector space.
Fundamental Subspaces Here we will take a look at some of the fundamental subspaces of a matrix, including the row space, column space and null space.
Inner Product Spaces We will be looking at a special kind of vector spaces in this section as well as define the inner product.
Orthonormal Basis In this section we will develop and use the Gram-Schmidt process for constructing an orthogonal/orthonormal basis for an inner product space.
Least Squares In this section we’ll take a look at an application of some of the ideas that we will be discussing in this chapter.
QR-Decomposition Here we will take a look at the QR-Decomposition for a matrix and how it can be used in the least squares process.
Orthogonal Matrices We will take a look at a special kind of matrix, the orthogonal matrix, in this section.
Eigenvalues and Eigenvectors
Review of Determinants In this section we’ll do a quick review of determinants.
Eigenvalues and Eigenvectors Here we will take a look at the main section in this chapter. We’ll be looking at the concept of Eigenvalues and Eigenvectors.
Diagonalization We’ll be looking at diagonalizable matrices in this section.
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This book helps students to master the material of a standard undergraduate linear algebra course.
The material is standard in that the topics covered are Gaussian reduction, vector spaces, linear maps, determinants, and eigenvalues and eigenvectors. The audience is also standard: sophmores or juniors, usually with a background of at least one semester of Calculus and perhaps with as much as three semesters.
The help that it gives to students comes from taking a developmental approach— this book’s presentation emphasizes motivation and naturalness, driven home by a wide variety of examples and extensive, careful, exercises. The developmental approach is what sets this book apart, so some expansion of the term is appropriate here.
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Numerical methods for solving large-scale linear algebraic problems are necessary to solve various partial/ordinary differential equations by the finite element method, finite difference method, spectral method, and etc. Also in the solution of optimization problems, such methods are essential.
Difficulties in practical computation of large linear systems arise from the following observations:
* usually the computational costs are too expensive;
* there will be possible loss in accuracy with a fixed number of digits computation;
* the methods are not applicable to different problems.
The main questions in numerical methods for linear systems are
* How fast is the numerical method in the sense of operation counts (flops: 1 flop=1 multiplication + 1 addition)?
* What is the accuracy? Can a priori and a posteriori estimates be given?
* What is the coverage of the method?
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This book is a survey of abstract algebra with emphasis on linear algebra. It is intended for students in mathematics, computer science, and the physical sciences. The first three or four chapters can stand alone as a one semester course in abstract algebra. However they are structured to provide the background for the chapter on linear algebra. Chapter 2 is the most difficult part of the book because groups are written in additive and multiplicative notation, and the concept of coset is confusing at first. After Chapter 2 the book gets easier as you go along. Indeed, after the first four chapters, the linear algebra follows easily. Finishing the chapter on linear algebra gives a basic one year undergraduate course in abstract algebra. Chapter 6 continues the material to complete a first year graduate course. Classes with little background can do the first three chapters in the first semester, and chapters 4 and 5 in the second semester. More advanced classes can do four chapters the first semester and chapters 5 and 6 the second semester. As bare as the first four chapters are, you still have to truck right along to finish them in one semester.
The presentation is compact and tightly organized, but still somewhat informal. The proofs of many of the elementary theorems are omitted. These proofs are to be provided by the professor in class or assigned as homework exercises. There is a non-trivial theorem stated without proof in Chapter 4, namely the determinant of the product is the product of the determinants. For the proper flow of the course, this theorem should be assumed there without proof. The proof is contained in Chapter 6. The Jordan form should not be considered part of Chapter 5. It is stated there only as a reference for undergraduate courses. Finally, Chapter 6 is not written primarily for reference, but as an additional chapter for more advanced courses.
This text is written with the conviction that it is more effective to teach abstract and linear algebra as one coherent discipline rather than as two separate ones. Teaching abstract algebra and linear algebra as distinct courses results in a loss of synergy and a loss of momentum. Also with this text the professor does not extract the course from the text, but rather builds the course upon it. I am convinced it is easier to build a course from a base than to extract it from a big book. Because after you extract it, you still have to build it. The bare bones nature of this book adds to its flexibility, because you can build whatever course you want around it. Basic algebra is a subject of incredible elegance and utility, but it requires a lot of organization. This book is my attempt at that organization. Every effort has been extended to make the subject move rapidly and to make the flow from one topic to the next as seamless as possible. The student has limited time during the semester for serious study, and this time should be allocated with care. The professor picks which topics to assign for serious study and which ones to "wave arms at". The goal is to stay focused and go forward, because mathematics is learned in hindsight.
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This book is an introduction to linear algebra, based on lectures given by me over 17 years, in the (now defunct) first year course MP103 at the University of Queensland.
The style is somewhat formal and terse, whereas in the lecture room I like to open up and present even the most boring things with enthusiasm and motivation.
The book Linear Algebra, an introduction with concurrent examples, by A.G. Hamilton, CUP 1989, is on a similar level, with much more emphasis on good pedagogy.
In the first edition I included a chapter on the LDU algorithm. However I omitted this from the second edition and instead recommend the reader to either C.G. Cullen, Linear Algebra with Applications or I.N. Herstein and D.J. Winter, A Primer of Linear Algebra, Macmillan 1988.
Students are encouraged to try the problems, which range from the mechanical to the more subtle, the latter demanding a greater level of interaction from the student.
The section on subspaces is meant to be a gentle introduction to the second course, where abstract vector spaces are met in detail. Things of substance are met here, including the rank of a matrix.
The section on three dimensional geometry makes use of the earlier sections on linear equations, matrices and determinants and some of the proofs are more algebraic (even pedantic) than some readers would like.
One criticism of the book has been its neglect of the computational side of the subject. This is partly a reflection of my love of discrete things such as integers, rational numbers and finite fields and a distrust of floating point arithmetic.
However, one redeeming feature is that I have written an exact arithmetic matrix program called CMAT, which performs exact calculations on matices whose elements are rational numbers, complex rational numbers or numbers from a finite field of p (prime) elements. CMAT takes the hard work out of calculating things such as the reduced row echelon form, the determinant and characteristic polynomial of a matrix.
Peter Adams produced the conics diagrams in Chapter 7 with his excellent CONICS program. Unfortunately this is not available in CMAT, as CONICS was written with specific graphics commands relevant to a special type of terminal!
I have also made the solutions to all problems in the notes available on the WWW.
The notes are freely available for educational purposes and are not to be used for monetary gain.
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This course ``Application of Linear Algebra'' is band on the lectures given of the late by the author to the postgraduate students at Tallinn Technical University. Our aim was to acquaint the students with the linear algebra packages LINPACK, EISPACK and LAPACK, and with the theoretical fundamentals of the parts of the packages MATLAB, MAPLE, MATHCAD and MATHEMATICA related to linear algebra. We have tried to explain the linear algebra methods which form the base for the computing methods used in the packages. We would like to stress that the aim of the course is not to work out concrete computing algorithms but to learn about the basic ideas related to these algorithms. It will be assumed that the reader is acquainted with the basic ideas of algebra.
The author would him to thank Assoc. Prof. Ellen Redi (Tallinn Pedagogical University) whose help by the improvement of the presented matherial both to its contents and its form has been enormous. Many of the given examples and problems were prepared by students Kristiina Krüspani, Kadri Mikk, Reena Prints (Tallinn Pedagogical University), Andrei Filonov, Dmitri Tseluiko (Tartu University) Juhan-Peep Ernits and Heiki Hiisjärv (Tallinn Technical University) within the framework of the TEMPUS-project during their stay at Tampere University of Technology in June, 1997.
The numbers of their examples and problems are marked by asterisk ``*''.
The matherial is band on the monographs of G.H.Golub and C.F.Van Loan (1996), and G.Strang (1988).
I hope that the course will help the reader interested in applications of linear algebra more consciously and effectively use the linear algebra packages.
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Here is a listing of all the material that is currently available online.
Preliminaries
Integer Exponents - In this section we will start looking at exponents and their properties.
Rational Exponents - We will define rational exponents in this section and extend the properties from the previous section to rational exponents.
Real Exponents - This is a short acknowledgment that the exponent properties form the previous two sections will hold for any exponent.
Radicals - Here we will define radical notation and relate radicals to rational exponents. We will also give the properties of radicals.
Polynomials - We will introduce the basics of polynomials in this section including adding, subtracting and multiplying polynomials.
Factoring Polynomials - This is the most important section of all the preliminaries. Factoring polynomials will appear in pretty much every chapter in this course. Without the ability to factor polynomials you will be unable to complete this course.
Rational Expressions - In this section we will define rational expressions and discuss adding, subtracting, multiplying and dividing them.
Complex Numbers - Here is a very quick primer on complex numbers and how to manipulate them.
Solving Equations and Inequalities
Solutions and Solution Sets - We introduce some of the basic notation and ideas involved in solving in this section.
Linear Equations - In this section we will solve linear equations, including equations with rational expressions.
Applications of Linear Equations - We will take a quick look at applications of linear equations in this section.
Equations With More Than One Variable - Here we will look at solving equations with more than one variable in them.
Quadratic Equations, Part I - In this section we will start looking at solving quadratic equations. We will look at factoring and the square root property in this section.
Quadratic Equations, Part II - We will finish up solving quadratic equations in this section. We will look at completing the square and quadratic formula in this section.
Quadratic Equations : A Summary - We’ll give a procedure for determining which method to use in solving quadratic equations in this section. We will also take a quick look at the discriminate.
Applications of Quadratic Equations - Here we will revisit some of the applications we saw in the linear application section, only this time they will involve solving a quadratic equation.
Equations Reducible to Quadratic Form - In this section we will solve equations that can be reduced to quadratic in form.
Equations with Radicals - Here we will solve equations with square roots in them.
Linear Inequalities - We will start solving inequalities in this section by looking at linear inequalities.
Polynomial Inequalities - In this section we will look at solving inequalities that contain polynomials.
Rational Inequalities Here we will solve inequalities involving rational expressions.
Absolute Value Equations - We will official define absolute value in this section and solve equation that contain absolute value.
Absolute Value Inequalities - We will solve inequalities that involve absolute value in this section.
Graphing and Functions
Graphing - In this section we will introduce the Cartesian coordinate system and most of the basics of graph equations.
Lines - Here we will review the main ideas from the study of lines including slope and the special forms of the equation of a line.
Circles - We will look at the equation of a circle and graphing circles in this section.
The Definition of a Function - We will discuss the definition of a function in this section. We will also introduce the idea of function evaluation.
Graphing Functions - In this section we will look at the basics of graphing functions. We will also graph some piecewise functions in this section.
Combining functions - Here we will look at basic arithmetic involving functions as well as function composition.
Inverse Functions - We will define and find inverse functions in this section.
Common Graphs
Lines, Circles and Piecewise Functions - This section is here only to acknowledge that we’ve already talked about graphing these in a previous chapter.
Parabolas - We’ll be graphing parabolas in this section.
Ellipses - In this section we will graph ellipses.
Hyperbolas - Here we will be graphing hyperbolas.
Miscellaneous Functions - In this section we will graph a couple of common functions that don’t really take all that much work to so. We’ll be looking at the constant function, square root, absolute value and a simple cubic function.
Transformations - We will be looking at shifts and reflections of graphs in this section. Collectively these are often called transformations.
Symmetry - We will briefly discuss the topic of symmetry in this section.
Rational Functions - In this section we will graph some rational functions. We will also be taking a look at vertical and horizontal asymptotes.
Polynomial Functions
Dividing Polynomials - We’ll review some of the basics of dividing polynomials in this section.
Zeroes/Roots of Polynomials - In this section we’ll define just what zeroes/roots of polynomials are and give some of the more important facts concerning them.
Graphing Polynomials - Here we will give a process that will allow us to get a rough sketch of some polynomials.
Finding Zeroes of Polynomials - We’ll look at a process that will allow us to find some of the zeroes of a polynomial and in special cases all of the zeroes.
Partial Fractions - In this section we will take a look at the process of partial fractions and finding the partial fraction decomposition of a rational expression.
Exponential and Logarithm Functions
Exponential Functions - In this section we will introduce exponential functions. We will be taking a look at some of the properties of exponential functions.
Logarithm Functions - Here we will introduce logarithm functions. We be looking at how to evaluate logarithms as well as the properties of logarithms.
Solving Exponential Equations - We will be solving equations that contain exponentials in this section.
Solving Logarithm Equations - Here we will solve equations that contain logarithms.
Applications - In this section we will look at a couple of applications of exponential functions and an application of logarithms.
Systems of Equations
Linear Systems with Two Variables - In this section we will use systems of two equations and two variables to introduce two of the main methods for solving systems of equations.
Linear Systems with Three Variables - Here we will work a quick example to show how to use the methods to solve systems of three equations with three variables.
Augmented Matrices - We will look at the third main method for solving systems in this section. We will look at systems of two equations and systems of three equations.
More on the Augmented Matrix - In this section we will take a look at some special cases to the solutions to systems and how to identify them using the augmented matrix method.
Nonlinear Systems - We will take a quick look at solving nonlinear systems of equations in this section.
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This text provides a thorough introduction to "modern'' or "abstract'' algebra at a level suitable for upper-level undergraduates and beginning graduate students. The book addresses the conventional topics: groups, rings, fields, and linear algebra, with symmetry as a unifying theme.
The most important goal of this book is to engage students in the active practice of mathematics. Students are given the opportunity to participate and investigate, starting on the first page. Exercises are plentiful, and working exercises should be the heart of the course.
The most important goal of this book is to engage students in the active practice of mathematics. Students are given the opportunity to participate and investigate, starting on the first page. Exercises are plentiful, and working exercises should be the heart of the course.
The required background for using this text is a standard first course in linear algebra. I have included a brief summary of linear algebra in an appendix to help students review. I have also provided appendices on sets, logic, mathematical induction, and complex numbers. It might also be useful to recommend a short supplementary text on set theory, logic, and proofs to be used as a reference and aid; several such texts are currently available.
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This is a text for the basic graduate sequence in abstract algebra, offered by most universities. We study fundamental algebraic structures, namely groups, rings, fields and
modules, and maps between these structures. The techniques are used in many areas of mathematics, and there are applications to physics, engineering and computer science as
well. In addition, I have attempted to communicate the intrinsic beauty of the subject. Ideally, the reasoning underlying each step of a proof should be completely clear, but the
overall argument should be as brief as possible, allowing a sharp overview of the result. These two requirements are in opposition, and it is my job as expositor to try to resolve the conflict.
My primary goal is to help the reader learn the subject, and there are times when informal or intuitive reasoning leads to greater understanding than a formal proof. In the
text, there are three types of informal arguments:
1. The concrete or numerical example with all features of the general case. Here, the
example indicates how the proof should go, and the formalization amounts to substituting Greek letters for numbers. There is no essential loss of rigor in the informal version.
2. Brief informal surveys of large areas. There are two of these, p-adic numbers and
group representation theory. References are given to books accessible to the beginning graduate student.
3. Intuitive arguments that replace lengthy formal proofs which do not reveal why a result is true. In this case, explicit references to a precise formalization are given. I am not saying that the formal proof should be avoided, just that the basic graduate year, where there are many pressing matters to cope with, may not be the appropriate place, especially when the result rather than the proof technique is used in applications.
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This site contains many of the definitions and theorems from the area of mathematics generally called abstract algebra. It is intended for undergraduate students taking an abstract algebra class at the junior/senior level, as well as for students taking their first graduate algebra course. It is based on the books Abstract Algebra, by John A. Beachy and William D. Blair, and Abstract Algebra II, by John A. Beachy.
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The original 1981 edition of A Course in Universal Algebra has now been LaTeXed so the authors could make the out-of-print Springer-Verlag Graduate Texts in Mathematics edition available once again, with corrections. The subject of Universal Algebra has flourished mightily since 1981, and we still believe that A Course in Universal Algebra offers an excellent introduction to the subject.
Universal algebra has enjoyed a particularly explosive growth in the last twenty years, and a student entering the subject now will find a bewildering amount of material to digest.
This text is not intended to be encyclopedic; rather, a few themes central to universal algebra have been developed sufficiently to bring the reader to the brink of current research. The choice of topics most certainly reflects the authors' interests.
Chapter I contains a brief but substantial introduction to lattices, and to the close connection between complete lattices and closure operators. In particular, everything necessary for the subsequent study of congruence lattices is included.
Chapter II develops the most general and fundamental notions of universal algebra| these include the results that apply to all types of algebras, such as the homomorphism and isomorphism theorems. Free algebras are discussed in great detail|we use them to derive the existence of simple algebras, the rules of equational logic, and the important Mal'cev conditions. We introduce the notion of classifying a variety by properties of (the lattices of) congruences on members of the variety. Also, the center of an algebra is defined and used to characterize modules (up to polynomial equivalence).
In Chapter III we show how neatly two famous results|the refutation of Euler's conjecture on orthogonal Latin squares and Kleene's characterization of languages accepted by finite automata - can be presented using universal algebra. We predict that such "applied universal algebra" will become much more prominent.
Chapter IV starts with a careful development of Boolean algebras, including Stone duality, which is subsequently used in our study of Boolean sheaf representations; however, the cumbersome formulation of general sheaf theory has been replaced by the considerably simpler definition of a Boolean product. First we look at Boolean powers, a beautiful tool for transferring results about Boolean algebras to other varieties as well as for providing a structure theory for certain varieties. The highlight of the chapter is the study of discriminator varieties. These varieties have played a remarkable role in the study of spectra, model companions, decidability, and Boolean product representations. Probably no other class of
varieties is so well-behaved yet so fascinating.
The final chapter gives the reader a leisurely introduction to some basic concepts, tools, and results of model theory. In particular, we use the ultraproduct construction to derive the compactness theorem and to prove fundamental preservation theorems. Principal congruence formulas are a favorite model-theoretic tool of universal algebraists, and we use them in the study of the sizes of subdirectly irreducible algebras. Next we prove three general results on the existence of a finite basis for an equational theory. The last topic is semantic embeddings, a popular technique for proving undecidability results. This technique is essentially algebraic in nature, requiring no familiarity whatsoever with the theory of algorithms. (The study of decidability has given surprisingly deep insight into the limitations of Boolean product representations.)
At the end of several sections the reader will find selected references to source material plus state of the art texts or papers relevant to that section, and at the end of the book one finds a brief survey of recent developments and several outstanding problems.
This is a text for a basic course in commutative algebra, written in accordance with the following objectives.
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The course should be accessible to those who have studied algebra at the beginning graduate level. For general algebraic background, see author's online text "Abstract Algebra: The Basic Graduate Year". This text will be referred to as TBGY.
The idea is to help the student reach an advanced level as quickly and efficiently as possible. In Chapter 1, the theory of primary decomposition is developed so as to apply to modules as well as ideals. In Chapter 2, integral extensions are treated in detail, including the lying over, going up and going down theorems. The proof of the going down theorem does not require advanced field theory. Valuation rings are studied in Chapter 3, and the characterization theorem for discrete valuation rings is proved. Chapter 4 discusses completion, and covers the Artin-Rees lemma and the Krull intersection theorem. Chapter 5 begins with a brief digression into the calculus of finite differences, which clarifies some of the manipulations involving Hilbert and Hilbert-Samuel polynomials. The main result is the dimension theorem for finitely generated modules over Noetherian local rings. A corollary is Krull’s principal ideal theorem. Some connections with algebraic geometry are established via the study of affine algebras. Chapter 6 introduces the fundamental notions of depth, systems of parameters, and M-sequences. Chapter 7 develops enough homological algebra to prove, under approprate hypotheses, that all maximal M-sequences have the same length. The brief Chapter 8 develops enough theory to prove that a regular local ring is an integral domain as well as a Cohen-Macaulay ring. After completing the course, the student should be equipped to meet the Koszul complex, the Auslander-Buchsbaum theorems, and further properties of Cohen-Macaulay rings in a more advanced course.
