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Linear Algebra Done Right
Sheldon Axler · Springer · 2024
Now available in Open Access, this best-selling textbook for a second course in linear algebra is aimed at undergraduate math majors and graduate students. The fourth edition gives an expanded treatment of the singular value decomposition and its consequences. It includes a new chapter on multilinear algebra, treating bilinear forms, quadratic forms, tensor products, and an approach to determinants via alternating multilinear forms. This new edition also increases the use of the minimal polynomial to provide cleaner proofs of multiple results. Also, over 250 new exercises have been added. The novel approach taken here banishes determinants to the end of the book. The text focuses on the central goal of linear algebra: understanding the structure of linear operators on finite-dimensional vector spaces. The author has taken unusual care to motivate concepts and simplify proofs. A variety of interesting exercises in each chapter helps students understand and manipulate the objects of linear algebra. Beautiful formatting creates pages with an unusually student-friendly appearance in both print and electronic versions. No prerequisites are assumed other than the usual demand for suitable mathematical maturity. The text starts by discussing vector spaces, linear independence, span, basis, and dimension. The book then deals with linear maps, eigenvalues, and eigenvectors. Inner-product spaces are introduced, leading to the finite-dimensional spectral theorem and its consequences. Generalized eigenvectors are then used to provide insight into the structure of a linear operator.
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Classical Numerical Methods in Scientific Computing
Jos van Kan; Guus Segal; Fred Vermolen; Hans Kraaijevanger · TU Delft Open · 2023
Partial differential equations are paramount in mathematical modelling with applications in engineering and science. The book starts with a crash course on partial differential equations in order to familiarize the reader with fundamental properties such as existence, uniqueness and possibly existing maximum principles. The main topic of the book entails the description of classical numerical methods that are used to approximate the solution of partial differential equations. The focus is on discretization methods such as the finite difference, finite volume and finite element method. The manuscript also makes a short excursion to the solution of large sets of (non)linear algebraic equations that result after application of discretization method to partial differential equations. The book treats the construction of such discretization methods, as well as some error analysis, where it is noted that the error analysis for the finite element method is merely descriptive, rather than rigorous from a mathematical point of view. The last chapters focus on time integration issues for classical time-dependent partial differential equations. After reading the book, the reader should be able to derive finite element methods, to implement the methods and to judge whether the obtained approximations are consistent with the solution to the partial differential equations. The reader will also obtain these skills for the other classical discretization methods. Acquiring such fundamental knowledge will allow the reader to continue studying more advanced methods like meshfree methods, discontinuous Galerkin methods and spectral methods for the approximation of solutions to partial differential equations.
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Contemporary Mathematics
Donna Kirk · OpenStax · 2023
Contemporary Mathematics is designed to meet the scope and sequence requirements for a liberal arts mathematics course. This resource provides stand-alone sections with a focus on showing relevance in the features as well as the examples, exercises, and exposition. Contemporary Mathematics integrates technology applications, projects, and highlights a diverse group of contributors to mathematics, statistics, and related fields. Print copies of this book will be available for the fall 2023 semester. You can sign up to be notified when print copies are available.
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COMPUTATIONAL PHYSICS: A Practical Introduction to Computational Physics and Scientific Computing (using C++)
Konstantinosn Anagnostopoulos · National Technical University of Athens · 2016
This book is an introduction to the computational methods used in physics, but also in other scientific fields. It is addressed to an audience that has already been exposed to the introductory level of college physics, usually taught during the first two years of an undergraduate program in science and engineering. It assumes no prior knowledge of numerical analysis, programming or computers and teaches whatever is necessary for the solution of the problems addressed in the text. It can be used as a textbook in introductory computational physics or scientific computing classes. The book starts with very simple problems in particle motion and ends with an in-depth discussion of advanced techniques used in Monte Carlo simulations in statistical mechanics. The level of instruction rises slowly, while discussing problems like the diffusion equation, electrostatics on the plane, quantum mechanics and random walks. All the material can be taught in two semesters, but a selection of topics can form the material of a one semester course. The book aims to provide the students with the background and the experience needed in order to advance to high performance computing projects in science and engineering. It puts emphasis on hands--on programming of numerical code but also on the production, analysis and interpretation of data. But it also tries to keep the students motivated by considering interesting applications in physics, like chaos, quantum mechanics, special relativity and the physics of phase transitions. There is a C++ and a Fortran edition for the core programming. Data analysis is performed using the powerful tools of the GNU/Linux environment. All the necessary software is open source and freely available. The book and the accompanying software are given under a Creative Commons License/GNU public License as a service to the community. It can be used freely as a whole, or any part of it, in any form, by anyone. There is no official distribution of hard copies, b
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Calculus in Context
James Callahan; David Cox; Kenneth Hoffman · Smith College Open Educational Resources: Textbooks · 1994
Designing the curriculum We believe that calculus can be for students what it was for Euler and the Bernoullis: a language and a tool for exploring the whole fabric of science. We also believe that much of the mathematical depth and vitality of calculus lies in connections to other sciences. The mathematical questions that arise are compelling in part because the answers matter to other disciplines. We began our work with a "clean slate," not by asking what parts of the traditional course to include or discard. Our starting points are thus our summary of what calculus is really about. Our curricular goals are what we aim to convey about the subject in the course. Our functional goals describe the attitudes and behaviors we hope our students will adopt in using calculus to approach scientific and mathematical questions. Starting Points Calculus is fundamentally a way of dealing with functional relationships that occur in scientific and mathematical contexts. The techniques of calculus must be subordinate to an overall view of the questions that give rise to these relationships. Technology radically enlarges the range of questions we can explore and the ways we can answer them. Computers and graphing calculators are much more than tools for teaching the traditional calculus. The concept of a dynamical system is central to science. Therefore, differential equations belong at the center of calculus, and technology makes this possible at the introductory level. The process of successive approximation is a key tool of calculus, even when the outcome of the process--the limit--cannot be explicitly given in closed form. Curricular Goals Develop calculus in the context of scientific and mathematical questions. Treat systems of differential equations as fundamental objects of study. Construct and analyze mathematical models. Use the method of successive approximations to define and solve problems. Develop geometric visualization with hand-drawn and computer graphics. Give num
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Discrete Mathematics: An Open Introduction
Oscar Levin · Oscar Levin · 2024
Discrete Mathematics: An Open Introduction is a free, open source textbook appropriate for a first or second year undergraduate course for math and computer science majors. The book is especially well-suited for courses that incorporate inquiry-based learning. Since Spring 2013, the book has been used as the primary textbook or a supplemental resource at more than 200 colleges and universities around the world (see the partial adoptions list). After many years of development, I am pleased to announce that the 4th edition of Discrete Mathematics: an Open Introduction is now available, here and on Runestone Academy. The new edition brings many improvements and a new organization of content. In particular, the book now starts with logic and proofs, then practices those proofs with graph theory. The second half of the book contains material on counting (with a new "application to probability" section) and sequences. Over the last few years, I have found that students have more success with this arrangement. There is also a stronger emphasis on discrete structures, which should make the book more useful for students in computer science, while still focusing on understanding mathematical concepts essential for math majors and future math teachers.
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Precalculus
Thomas Tradler; Holly Carley · CUNY Academic Works · 2015
These are notes for a course in precalculus, as it is taught at New York City College of Technology - CUNY (where it is offered under the course number MAT 1375). Our approach is calculator based. For this, we will use the currently standard TI-84 calculator, and in particular, many of the examples will be explained and solved with it. However, we want to point out that there are also many other calculators that are suitable for the purpose of this course and many of these alternatives have similar functionalities as the calculator that we have chosen to use. An introduction to the TI-84 calculator together with the most common applications needed for this course is provided in appendix A. In the future we may expand on this by providing introductions to other calculators or computer algebra systems. This course in precalculus has the overarching theme of “functions.” This means that many of the often more algebraic topics studied in the previous courses are revisited under this new function theoretic point of view. However, in order to keep this text as self contained as possible we always recall all results that are necessary to follow the core of the course even if we assume that the student has familiarity with the formula or topic at hand. After a first introduction to the abstract notion of a function, we study polynomials, rational functions, exponential functions, logarithmic functions, and trigonometric functions with the function viewpoint. Throughout, we will always place particular importance to the corresponding graph of the discussed function which will be analyzed with the help of the TI-84 calculator as mentioned above. These are in fact the topics of the first four (of the five) parts of this precalculus course. In the fifth and last part of the book, we deviate from the above theme and collect more algebraically oriented topics that will be needed in calculus or other advanced mathematics courses or even other science courses. This part includes a
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Introduction to Mathematical Analysis I
Beatriz Lafferriere; Gerardo Lafferriere; Mau Nam Nguyen · Portland State University Library · 2016
Our goal with this textbook is to provide students with a strong foundation in mathematical analysis. Such a foundation is crucial for future study of deeper topics of analysis. Students should be familiar with most of the concepts presented here after completing the calculus sequence. However, these concepts will be reinforced through rigorous proofs. The lecture notes contain topics of real analysis usually covered in a 10-week course: the completeness axiom, sequences and convergence, continuity, and differentiation. The lecture notes also contain many well-selected exercises of various levels. Although these topics are written in a more abstract way compared with those available in some textbooks, teachers can choose to simplify them depending on the background of the students. For instance, rather than introducing the topology of the real line to students, related topological concepts can be replaced by more familiar concepts such as open and closed intervals. Some other topics such as lower and upper semicontinuity, differentiation of convex functions, and generalized differentiation of non-differentiable convex functions can be used as optional mathematical projects. In this way, the lecture notes are suitable for teaching students of different backgrounds. The second edition includes a number of improvements based on recommendations from students and colleagues and on our own experience teaching the course over the last several years. In this edition we streamlined the narrative in several sections, added more proofs, many examples worked out in detail, and numerous new exercises. In all we added over 50 examples in the main text and 100 exercises (counting parts).
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Abstract Algebra: Theory and Applications
Thomas Judson · University of Puget Sound · 2016
This text is intended for a one- or two-semester undergraduate course in abstract algebra. Traditionally, these courses have covered the theoretical aspects of groups, rings, and fields. However, with the development of computing in the last several decades, applications that involve abstract algebra and discrete mathematics have become increasingly important, and many science, engineering, and computer science students are now electing to minor in mathematics. Though theory still occupies a central role in the subject of abstract algebra and no student should go through such a course without a good notion of what a proof is, the importance of applications such as coding theory and cryptography has grown significantly. Until recently most abstract algebra texts included few if any applications. However, one of the major problems in teaching an abstract algebra course is that for many students it is their first encounter with an environment that requires them to do rigorous proofs. Such students often find it hard to see the use of learning to prove theorems and propositions; applied examples help the instructor provide motivation. This text contains more material than can possibly be covered in a single semester. Certainly there is adequate material for a two-semester course, and perhaps more; however, for a one-semester course it would be quite easy to omit selected chapters and still have a useful text. The order of presentation of topics is standard: groups, then rings, and finally fields. Emphasis can be placed either on theory or on applications. A typical one-semester course might cover groups and rings while briefly touching on field theory, using Chapters 1 through 6, 9, 10, 11, 13 (the first part), 16, 17, 18 (the first part), 20, and 21. Parts of these chapters could be deleted and applications substituted according to the interests of the students and the instructor. A two-semester course emphasizing theory might cover Chapters 1 through 6, 9, 10, 11, 13
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How We Got from There to Here: A Story of Real Analysis
Robert Rogers; Eugene Boman · Open SUNY · 2014
The typical introductory real analysis text starts with an analysis of the real number system and uses this to develop the definition of a limit, which is then used as a foundation for the definitions encountered thereafter. While this is certainly a reasonable approach from a logical point of view, it is not how the subject evolved, nor is it necessarily the best way to introduce students to the rigorous but highly non-intuitive definitions and proofs found in analysis. This book proposes that an effective way to motivate these definitions is to tell one of the stories (there are many) of the historical development of the subject, from its intuitive beginnings to modern rigor. The definitions and techniques are motivated by the actual difficulties encountered by the intuitive approach and are presented in their historical context. However, this is not a history of analysis book. It is an introductory analysis textbook, presented through the lens of history. As such, it does not simply insert historical snippets to supplement the material. The history is an integral part of the topic, and students are asked to solve problems that occur as they arise in their historical context. This book covers the major topics typically addressed in an introductory undergraduate course in real analysis in their historical order. Written with the student in mind, the book provides guidance for transforming an intuitive understanding into rigorous mathematical arguments. For example, in addition to more traditional problems, major theorems are often stated and a proof is outlined. The student is then asked to fill in the missing details as a homework problem.
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Intermediate Algebra
John Redden · Saylor Foundation · 2012
It is essential to lay a solid foundation in mathematics if a student is to be competitive in today's global market. The importance of algebra, in particular, cannot be overstated, as it is the basis of all mathematical modeling used in applications found in all disciplines. Traditionally, the study of algebra is separated into a two parts, Elementary and Intermediate Algebra. This textbook by John Redden, Intermediate Algebra, is the second part. Written in a clear and concise manner, it carefully builds on the basics learned in Elementary Algebra and introduces the more advanced topics required for further study in applications found in most disciplines. Used as a standalone textbook, Intermediate Algebra offers plenty of review as well as something new to engage the student in each chapter. Written as a blend of the traditional and graphical approaches to the subject, this textbook introduces functions early and stresses the geometry behind the algebra. While CAS independent, a standard scientific calculator will be required and further research using technology is encouraged. Intermediate Algebra is written from the ground up in an open and modular format, allowing the instructor to modify it and leverage their individual expertise as a means to maximize the student experience and success. A more modernized element, embedded video examples, are present, but the importance of practice with pencil and paper is consistently stressed. Therefore, this text respects the traditional approaches to algebra pedagogy while enhancing it with the technology available today. The importance of Algebra cannot be overstated; it is the basis for all mathematical modeling used in all disciplines. After completing a course sequence based on Elementary and Intermediate Algebra, students will be on firm footing for success in higher-level studies at the college level.
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Math in Society
David Lippman · David Lippman · 2017
Math in Society is a free, open textbook. This book is a survey of contemporary mathematical topics, most non-algebraic, appropriate for a college-level topics course for liberal arts majors. The text is designed so that most chapters are independent, allowing the instructor to choose a selection of topics to be covered. Emphasis is placed on the applicability of the mathematics. Core material for each topic is covered in the main text, with additional depth available through exploration exercises appropriate for in-class, group, or individual investigation. This book is appropriate for Math 107 (Washington State Community Colleges common course number).
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Elementary Algebra
John Redden · Saylor Foundation · 2011
It is essential to lay a solid foundation in mathematics if a student is to be competitive in today's global market. The importance of algebra, in particular, cannot be overstated, as it is the basis of all mathematical modeling used in applications found in all disciplines. Traditionally, the study of algebra is separated into a two parts, elementary algebra and intermediate algebra. This textbook, Elementary Algebra, is the first part, written in a clear and concise manner, making no assumption of prior algebra experience. It carefully guides students from the basics to the more advanced techniques required to be successful in the next course. This text is, by far, the best elementary algebra textbook offered under a Creative Commons license. It is written in such a way as to maintain maximum flexibility and usability. A modular format was carefully integrated into the design. For example, certain topics, like functions, can be covered or omitted without compromising the overall flow of the text. An introduction of square roots in Chapter 1 is another example that allows for instructors wishing to include the quadratic formula early to do so. Topics such as these are carefully included to enhance the flexibility throughout. This textbook will effectively enable traditional or nontraditional approaches to elementary algebra. This, in addition to robust and diverse exercise sets, provides the base for an excellent individualized textbook instructors can use free of needless edition changes and excessive costs! A few other differences are highlighted below: Equivalent mathematical notation using standard text found on a keyboard A variety of applications and word problems included in most exercise sets Clearly enumerated steps found in context within carefully chosen examples Alternative methods and notation, modularly integrated, where appropriate Video examples available, in context, within the online version of the textbook Robust and diverse exercise sets with di
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Tort Law: Cases and Commentaries
Samuel Beswick · CanLII: Canadian Legal Information Institute · 2021
The law of obligations concerns the legal rights and duties owed between people. Three primary categories make up the common law of obligations: tort, contract, and unjust enrichment. This casebook provides an introduction to tort law: the law that recognises and responds to civil wrongdoing. The material is arranged in two parts. Part I comprises §§1-11 and addresses intentional and dignitary torts and the overarching theories and goals of tort law. Part II comprises §§12-25 and addresses no-fault compensation schemes, negligence, nuisance, strict liability, and tort law’s place within our broader legal systems. This casebook was compiled and edited by Assistant Professor Samuel Beswick of the University of British Columbia Peter A. Allard School of Law. Maddison Zapach (J.D. expected 2023) provided research assistance on the first edition (published July 2021). We gratefully acknowledge the influence on our approach to this subject of Professor Joost Blom QC of the Allard School of Law, Professor John C.P. Goldberg of Harvard Law School, and Associate Professor Rosemary Tobin of the University of Auckland Faculty of Law. The support of Open UBC and the UBC Teaching and Learning Enhancement Fund is also gratefully acknowledged.
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Introduction to business law in Papua New Guinea
Andrew Gibson · Southern Cross University · 2024
“Introduction to Business Law in Papua New Guinea” is an introductory text designed to support students enrolled in LEGL1007 Introduction to the Business Law of Papua New Guinea at Southern Cross University. The book is structured into five parts: Part 1 provides an introduction to studying business law, including writing legal essays and problem questions; Part 2 covers negligence; Part 3 discusses contract law; Part 4 addresses consumer law; and Part 5 focuses on agency within the Papua New Guinea legal system.
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Legal Writing I & II: Legal Research and Writing & Introduction to Litigation Practice
Ben Fernandez · Ben Fernandez · 2020
Legal Writing I & II; Legal Research and Writing & Introduction to Litigation Practice contains a brief discussion of all of the topics covered in a law school courses on legal writing, including a typical first semester course on legal research, analysis and writing an objective memorandum, as well as a second semester course on persuasion and writing an appellate brief, motion to dismiss or motion for summary judgment. The discussion focuses on the basics of analogical reasoning and persuasion and leaves out the minutiae. Each topic is taken one step at a time, with each step building on the step before it. The sources of law are presented first, then legal research, and reading and analyzing cases and statutes. The book covers analogizing a case to a fact pattern and marshaling the relevant facts to the elements of a statutory rule next. And then first section of the book concludes with legal citation, CRAC and CREAC, and writing a legal research memorandum. The text also includes a lot of samples and examples of how the author would write a case brief, a legal memoranda and an appellate brief, as well as an appendix with charts, outlines and exercises students can use to practice these skills. Legal Writing I & II; Legal Research and Writing & Introduction to Litigation Practice covers all the skills students need to know to work at a law firm, and everything students have to learn to begin practicing in litigation department of a firm.
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Sexual Orientation and Gender Identity Employment Discrimination
Matthew Green · CALI's eLangdell® Press · 2017
This Chapter will address the current protections that are available to lesbian, gay, bisexual and transgender (“LGBT”) individuals who allege they have been victims of employment discrimination. The Chapter's primary focus will be on federal statutory law, particularly Title VII of the Civil Rights Act of 1964. Although the focus here is on federal law, Appendix I to this Chapter lists the states that protect individuals from public and/or private discrimination under state laws. This topic is explored in four parts: (1) a brief overview of congressional efforts to enact a statute to protect individuals from employment discrimination on the basis of sexual orientation and gender identity; (2) discusses Title VII and sexual orientation; (3) discusses ways in which recent courts have handled sexual orientation discrimination under Title VII; and (4) similarly examines early judicial treatment of claims brought by individuals alleging discrimination on the basis of their gender identity and/or expression and explores how the law has developed in this area as well.
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U.S. Federal Income Taxation of Individuals 2023
Deborah Geier · CALI's eLangdell® Press · 2017
This is the tenth edition of this textbook, updated through December 15, 2022 for use beginning January 2023. In addition to incorporating new law and all inflation adjustments, this edition builds on updates made for the ninth edition and incorporates new charts pertaining to economic and tax data, including December 2019 CBO charts showing that income inequality between 2015 and 2020 (with the Tax Cuts and Jobs Act enacted in 2017) worsened after taxes and transfers than before taxes and transfers are taken into account. This textbook is not intended to be an exhaustive treatise; rather, it is intended to be far more useful than that for beginning tax law students by equipping the novice not merely with unmoored detail but rather with a rich blueprint that illuminates the deeper structural framework on which that detail hangs (sometimes crookedly). Chapter 1 outlines the conceptual meaning of the term income for uniquely tax purposes and examines the Internal Revenue Code provisions that translate this construct into positive law. Chapter 2 explores various forms of consumption taxation because the modern Internal Revenue Code is best perceived as a hybrid income-consumption tax that also contains many provisions that are inconsistent with both forms of taxation. Chapter 3 then provides students with the story of how we got to where we are today, important context about the distribution of the tax burden, the budget, and economic trends, as well as material on ethical debates, economic theories, and politics as they affect taxation. Armed with this larger blueprint, students are in a much better position to evaluate the myriad pieces that follow throughout the remaining 18 chapters. For example, they are in a better position to appreciate how applying the income tax rules for debt to a debt-financed investment afforded more favorable consumption tax treatment creates tax shelter problems. Stated another way, they can better appreciate how the tax system can someti
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Sales and Leases: A Problem-based Approach
Scott Burnham; Kristen Juras · CALI's eLangdell® Press · 2023
Sales and Leases is a coursebook for a 3-credit course in personal property sales and leases – the subject matter of UCC Articles 2 and 2A. Adjustments could be made for other credit allocations and chapters can be used on a stand-alone basis. The course is designed so that students both review the rules and principles they studied in their first-year course in Contracts and learn the rules that apply to the subset of contracts for the sale and lease of goods. Students taking this course should be well-prepared to solve legal problems in contracts and sales, and should be well-prepared for those parts of the bar exam as well. While the course in Contracts focuses on the rules of common-law contracts, the focus of this course is the rules found in legislation. Therefore, instead of emphasizing case analysis, the book contains a good deal of narrative that assists students in working through the complexity of the statutes. Students will need to supplement the book with a complete copy of the UCC that includes the Official Comments. Discussion of other statutes, such as UETA and Magnuson-Moss, is incorporated where appropriate. The approach is problem-based, which we believe is more appropriate for an upper-division course based primarily on statutes. The narrative is interspersed with problems for class discussion that require students to apply the principles and rules to particular fact situations. Many times there is an issue of interpretation or policy in the Code, however, and therefore each chapter also contains at least one case that explores an issue arising under the statute.
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Law of Wills
Browne Lewis · CALI's eLangdell® Press · 2016
The purpose of this casebook is to train law students to think and act like probate attorneys. This book is meant to be used in conjunction with the author's book on the law of trusts. This book's focus is problem-solving and legal application; the book includes numerous problems, so law students can learn to apply the law they learn from reading the cases. It also contains collaborative learning exercises to encourage students to engage in group problem-solving. The book is divided into three parts to reflect the main types of issues that students will encounter if they practice probate law. The book's organization mirrors the manner in which probate law is practiced in the real world. The book starts with an examination of the intestacy system because the majority of people die without executing a will. Therefore, most of the legal issues a probate lawyer faces center around the intestacy system. Unlike the typical wills casebook, this book provides a detailed discussion of the intestacy system. A chapter on ethics is included because probate attorneys encounter ethical issues that are different from attorneys practicing in other areas of law. The second part of the book includes an exploration of the testacy system. It is arranged so professors can lead students from the client interview to the will execution. The first three chapters of this section deal with issues that directly impact the existence of the inheritance system. It analyzes a person's ability to control the disposition of his or her property after death. This serves as the students' first introduction to the power of the “dead hand”. These chapters are included to start a public policy discussion about the rights of the dead, the right of heirs, and the necessity of an inheritance system. I tell my students that, when executing a will, they must think of the ways that it can be contested. In addition, I tell them that a will can be contested on two fronts-an attack on the testator and an attack on