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Beezer University of Puget Sound http://buzzard.ups.edu/ To my wife, Pat. 20042013 Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the appendix entitled GNU Free Documentation License. The most recent version can always be found at . Aidan Meacham Robert A. Beezer
Congruent Press
Gig Harbor, Washington, USA

Robert A. Beezer is a Professor of Mathematics at the University of Puget Sound, where he has been on the faculty since 1984. He received a B.S. in Mathematics (with an Emphasis in Computer Science) from the University of Santa Clara in 1978, a M.S. in Statistics from the University of Illinois at Urbana-Champaign in 1982 and a Ph.D. in Mathematics from the University of Illinois at Urbana-Champaign in 1984.

In addition to his teaching at the University of Puget Sound, he has made sabbatical visits to the University of the West Indies (Trinidad campus) and the University of Western Australia. He has also given several courses in the Master's program at the African Institute for Mathematical Sciences, South Africa. He has been a Sage developer since 2008.

He teaches calculus, linear algebra and abstract algebra regularly, while his research interests include the applications of linear algebra to graph theory. His professional website is at \url{http://buzzard.ups.edu}.

David Beezer Santa Clara U. Robert Beezer U. of Puget Sound buzzard.ups.edu/ Chris Black David Braithwaite Chicago, Illinois Sara Bucht U. of Puget Sound Steve Canfield U. of Puget Sound Dupont Hubert Creteil, France Sarah Fellez U. of Puget Sound Eric Fickenscher U. of Puget Sound Martin Jackson U. of Puget Sound www.math.ups.edu/~martinj Ivan Kessler U. of Puget Sound Don Kreher Michigan Tech. U. www.math.mtu.edu/~kreher Mark Hamrick St. Louis U. Jacob Linenthal U. of Puget Sound Elizabeth Million U. of Puget Sound Travis Osborne U. of Puget Sound Joe Riegsecker Middlebury, Indiana joepye (at) pobox (dot) com Manley Perkel U. of Puget Sound Douglas Phelps U. of Puget Sound Mark Shoemaker U. of Puget Sound Zoltan Toth zoli.web.elte.hu Tyler Ueltschi U. of Puget Sound Andy Zimmer U. of Puget Sound

Many people have helped to make this book, and its freedoms, possible.

First, the time to create, edit and distribute the book has been provided implicitly and explicitly by the University of Puget Sound. A sabbatical leave Spring 2004, a course release in Spring 2007 and a Lantz Senior Fellowship for the 2010-11 academic year are three obvious examples of explicit support. The course release was provided by support from the Lind-VanEnkevort Fund. The university has also provided clerical support, computer hardware, network servers and bandwidth. Thanks to Dean Kris Bartanen and the chairs of the Mathematics and Computer Science Department, Professors Martin Jackson, Sigrun Bodine and Bryan Smith, for their support, encouragement and flexibility.

My colleagues in the Mathematics and Computer Science Department have graciously taught our introductory linear algebra course using earlier versions and have provided valuable suggestions that have improved the book immeasurably. Thanks to Professor Martin Jackson (v0.30), Professor David Scott (v0.70), Professor Bryan Smith (v0.70, 0.80, v1.00, v2.00, v2.20), Professor Manley Perkel (v2.10), and Professor Cynthia Gibson (v2.20).

University of Puget Sound librarians Lori Ricigliano, Elizabeth Knight and Jeanne Kimura provided valuable advice on production, and interesting conversations about copyrights.

Many aspects of the book have been influenced by insightful questions and creative suggestions from the students who have labored through the book in our courses. For example, the flashcards with theorems and definitions are a direct result of a student suggestion. I will single out a handful of students at the University of Puget Sound who have been especially adept at finding and reporting mathematically significant typographical errors: Jake Linenthal, Christie Su, Kim Le, Sarah McQuate, Andy Zimmer, Travis Osborne, Andrew Tapay, Mark Shoemaker, Tasha Underhill, Tim Zitzer, Elizabeth Million, Steve Canfield, Jinshil Yi, Cliff Berger, Preston Van Buren, Duncan Bennett, Dan Messenger, Caden Robinson, Glenna Toomey, Tyler Ueltschi and Kyle Whitcomb. All the students of the Fall~2012 Math 290 sections were very helpful and patient through the major changes required in making Version 3.00.

I have tried to be as original as possible in the organization and presentation of this beautiful subject. However, I have been influenced by many years of teaching from another excellent textbook, Introduction to Linear Algebra by L.W. Johnson, R.D. Reiss and J.T. Arnold. When I have needed inspiration for the correct approach to particularly important proofs, I have learned to eventually consult two other textbooks. Sheldon Axler's Linear Algebra Done Right is a highly original exposition, while Ben Noble's Applied Linear Algebra frequently strikes just the right note between rigor and intuition. Noble's excellent book is highly recommended, even though its publication dates to 1969.

Conversion to various electronic formats have greatly depended on assistance from: Eitan Gurari, author of the powerful LaTeX translator, tex4ht; Davide Cervone, author of jsMath and MathJax; and Carl Witty, who advised and tested the Sony Reader format. Thanks to these individuals for their critical assistance.

Incorporation of Sage code is made possible by the entire community of Sage developers and users, who create and refine the mathematical routines, the user interfaces and applications in educational settings. Technical and logistical aspects of incorporating Sage code in open textbooks was supported by a grant from the United States National Science Foundation (DUE-1022574), which has been administered by the American Institute of Mathematics, and in particular, David Farmer. The support and assistance of my fellow Principal Investigators, Jason Grout, Tom Judson, Kiran Kedlaya, Sandra Laursen, Susan Lynds, and William Stein is especially appreciated.

David Farmer and Sally Koutsoliotas are responsible for the vision and initial experiments which lead to the knowl-enabled web version, as part of the Version 3 project.

General support and encouragement of free and affordable textbooks, in addition to specific promotion of this text, was provided by Nicole Allen, Textbook Advocate at Student Public Interest Research Groups. Nicole was an early consumer of this material, back when it looked more like lecture notes than a textbook.

Finally, in every respect, the production and distribution of this book has been accomplished with open source software. The range of individuals and projects is far too great to pretend to list them all. This project is an attempt to pay it forward.

This text is designed to teach the concepts and techniques of basic linear algebra as a rigorous mathematical subject. Besides computational proficiency, there is an emphasis on understanding definitions and theorems, as well as reading, understanding and creating proofs. A strictly logical organization, complete and exceedingly detailed proofs of every theorem, advice on techniques for reading and writing proofs, and a selection of challenging theoretical exercises will slowly provide the novice with the tools and confidence to be able to study other mathematical topics in a rigorous fashion.

Most students taking a course in linear algebra will have completed courses in differential and integral calculus, and maybe also multivariate calculus, and will typically be second-year students in university. This level of mathematical maturity is expected, however there is little or no requirement to know calculus itself to use this book successfully. With complete details for every proof, for nearly every example, and for solutions to a majority of the exercises, the book is ideal for self-study, for those of any age.

While there is an abundance of guidance in the use of the software system, , there is no attempt to address the problems of numerical linear algebra, which are arguably continuous in nature. Similarly, there is little emphasis on a geometric approach to problems of linear algebra. While this may contradict the experience of many experienced mathematicians, the approach here is consciously algebraic. As a result, the student should be well-prepared to encounter groups, rings and fields in future courses in algebra, or other areas of discrete mathematics.

How to Use This Book

While the book is divided into chapters, the main organizational unit is the thirty-seven sections. Each contains a selection of definitions, theorems, and examples interspersed with commentary. If you are enrolled in a course, read the section before class and then answer the section's reading questions as preparation for class.

The version available for viewing in a web browser is the most complete, integrating all of the components of the book. Consider acquainting yourself with this version. Knowls are indicated by a dashed underlines and will allow you to seamlessly remind yourself of the content of definitions, theorems, examples, exercises, subsections and more. Use them liberally.

Historically, mathematics texts have numbered definitions and theorems. We have instead adopted a strategy more appropriate to the heavy cross-referencing, linking and knowling afforded by modern media. Mimicking an approach taken by Donald Knuth, we have given items short titles and associated acronyms. You will become comfortable with this scheme after a short time, and might even come to appreciate its inherent advantages. In the web version, each chapter has a list of ten or so important items from that chapter, and you will find yourself recognizing some of these acronyms with no extra effort beyond the normal amount of study.

Exercises come in three flavors, indicated by the first letter of their label. C indicates a problem that is essentially computational. T represents a problem that is more theoretical, usually requiring a solution that is as rigorous as a proof. M stands for problems that are medium, moderate, midway, mediate or median, but never mediocre. Their statements could feel computational, but their solutions require a more thorough understanding of the concepts or theory, while perhaps not being as rigorous as a proof. Of course, such a tripartite division will be subject to interpretation. Otherwise, larger numerical values indicate greater perceived difficulty, with gaps allowing for the contribution of new problems from readers. Many, but not all, exercises have complete solutions. These are indicated by daggers in the PDF and print versions, with solutions available in an online supplement, while in the web version a solution is indicated by a knowl right after the problem statement. Resist the urge to peek early. Working the exercises diligently is the best way to master the material.

The Archetypes are a collection of twenty-four archetypical examples. The open source lexical database, WordNet, defines an archetype as something that serves as a model or a basis for making copies. We employ the word in the first sense here. By carefully choosing the examples we hope to provide at least one example that is interesting and appropriate for many of the theorems and definitions, and also provide counterexamples to conjectures (and especially counterexamples to converses of theorems). Each archetype has numerous computational results which you could strive to duplicate as you encounter new definitions and theorems. There are some exercises which will help guide you in this quest.

Supplements

Print versions of the book (either a physical copy or a PDF version) have significant material available as supplements. Solutions are contained in the Exercise Manual. Advice on the use of the open source mathematical software system, , is contained in another supplement. (Look for a linear algebra Quick Reference sheet at the website.) The Archetypes are available in a PDF form which could be used as a workbook. Flashcards, with the statement of every definition and theorem, in order of appearance, are also available.

Freedom

This book is copyrighted by its author. Some would say it is his intellectual property, a distasteful phrase if there ever was one. Rather than exercise all the restrictions provided by the government-granted monopoly that is copyright, the author has granted you a license, the . In summary it says you may receive an electronic copy at no cost via electronic networks and you may make copies forever. So your copy of the book never has to go out-of-print. You may redistribute copies and you may make changes to your copy for your own use. However, you have one major responsibility in accepting this license. If you make changes and distribute the changed version, then you must offer the same license for the new version, you must acknowledge the original author's work, and you must indicate where you have made changes.

In practice, if you see a change that needs to be made (like correcting an error, or adding a particularly nice theoretical exercise), you may just wish to donate the change to the author rather than create and maintain a new version. Such donations are highly encouraged and gratefully accepted. You may notice the large number of small mistakes that have been corrected by readers that have come before you. Pay it forward.

So, in one word, the book really is free (as in no cost). But the open license employed is vastly different than free to download, all rights reserved. Most importantly, you know that this book, and its ideas, are not the property of anyone. Or they are the property of everyone. Either way, this book has its own inherent freedom, separate from those who contribute to it. Much of this philosophy is embodied in the following quote:

If nature has made any one thing less susceptible than all others of exclusive property, it is the action of the thinking power called an idea, which an individual may exclusively possess as long as he keeps it to himself; but the moment it is divulged, it forces itself into the possession of every one, and the receiver cannot dispossess himself of it. Its peculiar character, too, is that no one possesses the less, because every other possesses the whole of it. He who receives an idea from me, receives instruction himself without lessening mine; as he who lights his taper at mine, receives light without darkening me. That ideas should freely spread from one to another over the globe, for the moral and mutual instruction of man, and improvement of his condition, seems to have been peculiarly and benevolently designed by nature, when she made them, like fire, expansible over all space, without lessening their density in any point, and like the air in which we breathe, move, and have our physical being, incapable of confinement or exclusive appropriation.

Thomas Jefferson
Letter to Isaac McPherson
August 13, 1813

To the Instructor

The first half of this text (through ) is a course in matrix algebra, though the foundation of some more advanced ideas is also being formed in these early sections (such as , which presages invertible linear transformations). Vectors are presented exclusively as column vectors (not transposes of row vectors), and linear combinations are presented very early. Spans, null spaces, column spaces and row spaces are also presented early, simply as sets, saving most of their vector space properties for later, so they are familiar objects before being scrutinized carefully.

You cannot do everything early, so in particular matrix multiplication comes later than usual. However, with a definition built on linear combinations of column vectors, it should seem more natural than the more frequent definition using dot products of rows with columns. And this delay emphasizes that linear algebra is built upon vector addition and scalar multiplication. Of course, matrix inverses must wait for matrix multiplication, but this does not prevent nonsingular matrices from occurring sooner. Vector space properties are hinted at when vector and matrix operations are first defined, but the notion of a vector space is saved for a more axiomatic treatment later (). Once bases and dimension have been explored in the context of vector spaces, linear transformations and their matrix representation follow. The predominant purpose of the book is the four sections of , which introduces the student to representations of vectors and matrices, change-of-basis, and orthonormal diagonalization (the spectral theorem). This final chapter pulls together all the important ideas of the previous chapters.

Our vector spaces use the complex numbers as the field of scalars. This avoids the fiction of complex eigenvalues being used to form scalar multiples of eigenvectors. The presence of the complex numbers in the earliest sections should not frighten students who need a review, since they will not be used heavily until much later, and provides a quick review.

Linear algebra is an ideal subject for the novice mathematics student to learn how to develop a subject precisely, with all the rigor mathematics requires. Unfortunately, much of this rigor seems to have escaped the standard calculus curriculum, so for many university students this is their first exposure to careful definitions and theorems, and the expectation that they fully understand them, to say nothing of the expectation that they become proficient in formulating their own proofs. We have tried to make this text as helpful as possible with this transition. Every definition is stated carefully, set apart from the text. Likewise, every theorem is carefully stated, and almost every one has a complete proof. Theorems usually have just one conclusion, so they can be referenced precisely later. Definitions and theorems are cataloged in order of their appearance ( and in the Reference chapter at the end of the book). Along the way, there are discussions of some more important ideas relating to formulating proofs (), which is partly advice and partly a primer on logic.

Collecting responses to the Reading Questions prior to covering material in class will require students to learn how to read the material. Sections are designed to be covered in a fifty-minute lecture. Later sections are longer, but as students become more proficient at reading the text, it is possible to survey these longer sections at the same pace. With solutions to many of the exercises, students may be given the freedom to work homework at their own pace and style (individually, in groups, with an instructor's help, etc.). To compensate and keep students from falling behind, I give an examination on each chapter.

is a powerful open source program for advanced mathematics. It is especially robust for linear algebra. We have included an abundance of material which will help the student (and instructor) learn how to use Sage for the study of linear algebra and how to understand linear algebra better with Sage. This material is tightly integrated with the web version of the book and will become even easier to use since the technology for interfaces to Sage continues to rapidly evolve. Sage is highly capable for mathematical research as well, and so should be a tool that students can use in subsequent courses and careers.

Conclusion

Linear algebra is a beautiful subject. I have enjoyed preparing this exposition and making it widely available. Much of my motivation for writing this book is captured by the sentiments expressed by H.M. Cundy and A.P. Rollet in their Preface to the First Edition of Mathematical Models (1952), especially the final sentence,

This book was born in the classroom, and arose from the spontaneous interest of a Mathematical Sixth in the construction of simple models. A desire to show that even in mathematics one could have fun led to an exhibition of the results and attracted considerable attention throughout the school. Since then the Sherborne collection has grown, ideas have come from many sources, and widespread interest has been shown. It seems therefore desirable to give permanent form to the lessons of experience so that others can benefit by them and be encouraged to undertake similar work.

Foremost, I hope that students find their time spent with this book profitable. I hope that instructors find it flexible enough to fit the needs of their course. You can always find the latest version, and keep current with any changes, at the book's website (). I appreciate receiving suggestions, corrections, and other comments, so please do contact me.

Robert A. Beezer
Tacoma, Washington
December 2012

Version 1.3, 3 November 2008

Copyright © 2000, 2001, 2002, 2007, 2008 Free Software Foundation, Inc. <http://fsf.org/>

Everyone is permitted to copy and distribute verbatim copies of this license document, but changing it is not allowed.

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If you publish printed copies (or copies in media that commonly have printed covers) of the Document, numbering more than 100, and the Document's license notice requires Cover Texts, you must enclose the copies in covers that carry, clearly and legibly, all these Cover Texts: Front-Cover Texts on the front cover, and Back-Cover Texts on the back cover. Both covers must also clearly and legibly identify you as the publisher of these copies. The front cover must present the full title with all words of the title equally prominent and visible. You may add other material on the covers in addition. Copying with changes limited to the covers, as long as they preserve the title of the Document and satisfy these conditions, can be treated as verbatim copying in other respects.

If the required texts for either cover are too voluminous to fit legibly, you should put the first ones listed (as many as fit reasonably) on the actual cover, and continue the rest onto adjacent pages.

If you publish or distribute Opaque copies of the Document numbering more than 100, you must either include a machine-readable Transparent copy along with each Opaque copy, or state in or with each Opaque copy a computer-network location from which the general network-using public has access to download using public-standard network protocols a complete Transparent copy of the Document, free of added material. If you use the latter option, you must take reasonably prudent steps, when you begin distribution of Opaque copies in quantity, to ensure that this Transparent copy will remain thus accessible at the stated location until at least one year after the last time you distribute an Opaque copy (directly or through your agents or retailers) of that edition to the public.

It is requested, but not required, that you contact the authors of the Document well before redistributing any large number of copies, to give them a chance to provide you with an updated version of the Document.

4. MODIFICATIONS

You may copy and distribute a Modified Version of the Document under the conditions of sections 2 and 3 above, provided that you release the Modified Version under precisely this License, with the Modified Version filling the role of the Document, thus licensing distribution and modification of the Modified Version to whoever possesses a copy of it. In addition, you must do these things in the Modified Version:

• A. Use in the Title Page (and on the covers, if any) a title distinct from that of the Document, and from those of previous versions (which should, if there were any, be listed in the History section of the Document). You may use the same title as a previous version if the original publisher of that version gives permission.
• B. List on the Title Page, as authors, one or more persons or entities responsible for authorship of the modifications in the Modified Version, together with at least five of the principal authors of the Document (all of its principal authors, if it has fewer than five), unless they release you from this requirement.
• C. State on the Title page the name of the publisher of the Modified Version, as the publisher.
• D. Preserve all the copyright notices of the Document.
• F. Include, immediately after the copyright notices, a license notice giving the public permission to use the Modified Version under the terms of this License, in the form shown in the Addendum below.
• G. Preserve in that license notice the full lists of Invariant Sections and required Cover Texts given in the Document's license notice.
• H. Include an unaltered copy of this License.
• I. Preserve the section Entitled "History", Preserve its Title, and add to it an item stating at least the title, year, new authors, and publisher of the Modified Version as given on the Title Page. If there is no section Entitled "History" in the Document, create one stating the title, year, authors, and publisher of the Document as given on its Title Page, then add an item describing the Modified Version as stated in the previous sentence.
• J. Preserve the network location, if any, given in the Document for public access to a Transparent copy of the Document, and likewise the network locations given in the Document for previous versions it was based on. These may be placed in the "History" section. You may omit a network location for a work that was published at least four years before the Document itself, or if the original publisher of the version it refers to gives permission.
• K. For any section Entitled "Acknowledgements" or "Dedications", Preserve the Title of the section, and preserve in the section all the substance and tone of each of the contributor acknowledgements and/or dedications given therein.
• L. Preserve all the Invariant Sections of the Document, unaltered in their text and in their titles. Section numbers or the equivalent are not considered part of the section titles.
• M. Delete any section Entitled "Endorsements". Such a section may not be included in the Modified Version.
• N. Do not retitle any existing section to be Entitled "Endorsements" or to conflict in title with any Invariant Section.
• O. Preserve any Warranty Disclaimers.

If the Modified Version includes new front-matter sections or appendices that qualify as Secondary Sections and contain no material copied from the Document, you may at your option designate some or all of these sections as invariant. To do this, add their titles to the list of Invariant Sections in the Modified Version's license notice. These titles must be distinct from any other section titles.

You may add a section Entitled "Endorsements", provided it contains nothing but endorsements of your Modified Version by various partiesfor example, statements of peer review or that the text has been approved by an organization as the authoritative definition of a standard.

You may add a passage of up to five words as a Front-Cover Text, and a passage of up to 25 words as a Back-Cover Text, to the end of the list of Cover Texts in the Modified Version. Only one passage of Front-Cover Text and one of Back-Cover Text may be added by (or through arrangements made by) any one entity. If the Document already includes a cover text for the same cover, previously added by you or by arrangement made by the same entity you are acting on behalf of, you may not add another; but you may replace the old one, on explicit permission from the previous publisher that added the old one.

The author(s) and publisher(s) of the Document do not by this License give permission to use their names for publicity for or to assert or imply endorsement of any Modified Version.

5. COMBINING DOCUMENTS

You may combine the Document with other documents released under this License, under the terms defined in section 4 above for modified versions, provided that you include in the combination all of the Invariant Sections of all of the original documents, unmodified, and list them all as Invariant Sections of your combined work in its license notice, and that you preserve all their Warranty Disclaimers.

The combined work need only contain one copy of this License, and multiple identical Invariant Sections may be replaced with a single copy. If there are multiple Invariant Sections with the same name but different contents, make the title of each such section unique by adding at the end of it, in parentheses, the name of the original author or publisher of that section if known, or else a unique number. Make the same adjustment to the section titles in the list of Invariant Sections in the license notice of the combined work.

In the combination, you must combine any sections Entitled "History" in the various original documents, forming one section Entitled "History"; likewise combine any sections Entitled "Acknowledgements", and any sections Entitled "Dedications". You must delete all sections Entitled "Endorsements".

6. COLLECTIONS OF DOCUMENTS

You may make a collection consisting of the Document and other documents released under this License, and replace the individual copies of this License in the various documents with a single copy that is included in the collection, provided that you follow the rules of this License for verbatim copying of each of the documents in all other respects.

You may extract a single document from such a collection, and distribute it individually under this License, provided you insert a copy of this License into the extracted document, and follow this License in all other respects regarding verbatim copying of that document.

7. AGGREGATION WITH INDEPENDENT WORKS

A compilation of the Document or its derivatives with other separate and independent documents or works, in or on a volume of a storage or distribution medium, is called an "aggregate" if the copyright resulting from the compilation is not used to limit the legal rights of the compilation's users beyond what the individual works permit. When the Document is included in an aggregate, this License does not apply to the other works in the aggregate which are not themselves derivative works of the Document.

If the Cover Text requirement of section 3 is applicable to these copies of the Document, then if the Document is less than one half of the entire aggregate, the Document's Cover Texts may be placed on covers that bracket the Document within the aggregate, or the electronic equivalent of covers if the Document is in electronic form. Otherwise they must appear on printed covers that bracket the whole aggregate.

8. TRANSLATION

Translation is considered a kind of modification, so you may distribute translations of the Document under the terms of section 4. Replacing Invariant Sections with translations requires special permission from their copyright holders, but you may include translations of some or all Invariant Sections in addition to the original versions of these Invariant Sections. You may include a translation of this License, and all the license notices in the Document, and any Warranty Disclaimers, provided that you also include the original English version of this License and the original versions of those notices and disclaimers. In case of a disagreement between the translation and the original version of this License or a notice or disclaimer, the original version will prevail.

If a section in the Document is Entitled "Acknowledgements", "Dedications", or "History", the requirement (section 4) to Preserve its Title (section 1) will typically require changing the actual title.

9. TERMINATION

You may not copy, modify, sublicense, or distribute the Document except as expressly provided under this License. Any attempt otherwise to copy, modify, sublicense, or distribute it is void, and will automatically terminate your rights under this License.

However, if you cease all violation of this License, then your license from a particular copyright holder is reinstated (a) provisionally, unless and until the copyright holder explicitly and finally terminates your license, and (b) permanently, if the copyright holder fails to notify you of the violation by some reasonable means prior to 60 days after the cessation.

Moreover, your license from a particular copyright holder is reinstated permanently if the copyright holder notifies you of the violation by some reasonable means, this is the first time you have received notice of violation of this License (for any work) from that copyright holder, and you cure the violation prior to 30 days after your receipt of the notice.

Termination of your rights under this section does not terminate the licenses of parties who have received copies or rights from you under this License. If your rights have been terminated and not permanently reinstated, receipt of a copy of some or all of the same material does not give you any rights to use it.

10. FUTURE REVISIONS OF THIS LICENSE

The Free Software Foundation may publish new, revised versions of the GNU Free Documentation License from time to time. Such new versions will be similar in spirit to the present version, but may differ in detail to address new problems or concerns. See http://www.gnu.org/copyleft/.

Each version of the License is given a distinguishing version number. If the Document specifies that a particular numbered version of this License "or any later version" applies to it, you have the option of following the terms and conditions either of that specified version or of any later version that has been published (not as a draft) by the Free Software Foundation. If the Document does not specify a version number of this License, you may choose any version ever published (not as a draft) by the Free Software Foundation. If the Document specifies that a proxy can decide which future versions of this License can be used, that proxy's public statement of acceptance of a version permanently authorizes you to choose that version for the Document.

11. RELICENSING

"Massive Multiauthor Collaboration Site" (or "MMC Site") means any World Wide Web server that publishes copyrightable works and also provides prominent facilities for anybody to edit those works. A public wiki that anybody can edit is an example of such a server. A "Massive Multiauthor Collaboration" (or "MMC") contained in the site means any set of copyrightable works thus published on the MMC site.

"Incorporate" means to publish or republish a Document, in whole or in part, as part of another Document.

An MMC is "eligible for relicensing" if it is licensed under this License, and if all works that were first published under this License somewhere other than this MMC, and subsequently incorporated in whole or in part into the MMC, (1) had no cover texts or invariant sections, and (2) were thus incorporated prior to November 1, 2008.

The operator of an MMC Site may republish an MMC contained in the site under CC-BY-SA on the same site at any time before August 1, 2009, provided the MMC is eligible for relicensing.

To use this License in a document you have written, include a copy of the License in the document and put the following copyright and license notices just after the title page:

Copyright (C)  YEAR  YOUR NAME.     Permission is granted to copy, distribute and/or modify this document     under the terms of the GNU Free Documentation License, Version 1.3     or any later version published by the Free Software Foundation;     with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts.     A copy of the license is included in the section entitled "GNU     Free Documentation License".

If you have Invariant Sections, Front-Cover Texts and Back-Cover Texts, replace the "with Texts." line with this:

with the Invariant Sections being LIST THEIR TITLES, with the     Front-Cover Texts being LIST, and with the Back-Cover Texts being LIST.

If you have Invariant Sections without Cover Texts, or some other combination of the three, merge those two alternatives to suit the situation.

If your document contains nontrivial examples of program code, we recommend releasing these examples in parallel under your choice of free software license, such as the GNU General Public License, to permit their use in free software.

]]> } \bigskip Everyone is permitted to copy and distribute verbatim copies of this license document, but changing it is not allowed. \end{center} \begin{center} {\bf\large Preamble} \end{center} The purpose of this License is to make a manual, textbook, or other functional and useful document free'' in the sense of freedom: to assure everyone the effective freedom to copy and redistribute it, with or without modifying it, either commercially or noncommercially. Secondarily, this License preserves for the author and publisher a way to get credit for their work, while not being considered responsible for modifications made by others. This License is a kind of copyleft'', which means that derivative works of the document must themselves be free in the same sense. It complements the GNU General Public License, which is a copyleft license designed for free software. We have designed this License in order to use it for manuals for free software, because free software needs free documentation: a free program should come with manuals providing the same freedoms that the software does. But this License is not limited to software manuals; it can be used for any textual work, regardless of subject matter or whether it is published as a printed book. We recommend this License principally for works whose purpose is instruction or reference. \begin{center} {\Large\bf 1. APPLICABILITY AND DEFINITIONS\par} %% \phantomsection %% \addcontentsline{toc}{section}{1. APPLICABILITY AND DEFINITIONS} \end{center} This License applies to any manual or other work, in any medium, that contains a notice placed by the copyright holder saying it can be distributed under the terms of this License. Such a notice grants a world-wide, royalty-free license, unlimited in duration, to use that work under the conditions stated herein. The \textbf{Document}'', below, refers to any such manual or work. Any member of the public is a licensee, and is addressed as \textbf{you}''. You accept the license if you copy, modify or distribute the work in a way requiring permission under copyright law. A \textbf{Modified Version}'' of the Document means any work containing the Document or a portion of it, either copied verbatim, or with modifications and/or translated into another language. A \textbf{Secondary Section}'' is a named appendix or a front-matter section of the Document that deals exclusively with the relationship of the publishers or authors of the Document to the Document's overall subject (or to related matters) and contains nothing that could fall directly within that overall subject. (Thus, if the Document is in part a textbook of mathematics, a Secondary Section may not explain any mathematics.) The relationship could be a matter of historical connection with the subject or with related matters, or of legal, commercial, philosophical, ethical or political position regarding them. The \textbf{Invariant Sections}'' are certain Secondary Sections whose titles are designated, as being those of Invariant Sections, in the notice that says that the Document is released under this License. If a section does not fit the above definition of Secondary then it is not allowed to be designated as Invariant. The Document may contain zero Invariant Sections. If the Document does not identify any Invariant Sections then there are none. The \textbf{Cover Texts}'' are certain short passages of text that are listed, as Front-Cover Texts or Back-Cover Texts, in the notice that says that the Document is released under this License. A Front-Cover Text may be at most 5 words, and a Back-Cover Text may be at most 25 words. A \textbf{Transparent}'' copy of the Document means a machine-readable copy, represented in a format whose specification is available to the general public, that is suitable for revising the document straightforwardly with generic text editors or (for images composed of pixels) generic paint programs or (for drawings) some widely available drawing editor, and that is suitable for input to text formatters or for automatic translation to a variety of formats suitable for input to text formatters. A copy made in an otherwise Transparent file format whose markup, or absence of markup, has been arranged to thwart or discourage subsequent modification by readers is not Transparent. An image format is not Transparent if used for any substantial amount of text. A copy that is not Transparent'' is called \textbf{Opaque}''. Examples of suitable formats for Transparent copies include plain ASCII without markup, Texinfo input format, LaTeX input format, SGML or XML using a publicly available DTD, and standard-conforming simple HTML, PostScript or PDF designed for human modification. Examples of transparent image formats include PNG, XCF and JPG. Opaque formats include proprietary formats that can be read and edited only by proprietary word processors, SGML or XML for which the DTD and/or processing tools are not generally available, and the machine-generated HTML, PostScript or PDF produced by some word processors for output purposes only. The \textbf{Title Page}'' means, for a printed book, the title page itself, plus such following pages as are needed to hold, legibly, the material this License requires to appear in the title page. For works in formats which do not have any title page as such, Title Page'' means the text near the most prominent appearance of the work's title, preceding the beginning of the body of the text. The \textbf{publisher}'' means any person or entity that distributes copies of the Document to the public. A section \textbf{Entitled XYZ}'' means a named subunit of the Document whose title either is precisely XYZ or contains XYZ in parentheses following text that translates XYZ in another language. (Here XYZ stands for a specific section name mentioned below, such as \textbf{Acknowledgements}'', \textbf{Dedications}'', \textbf{Endorsements}'', or \textbf{History}''.) To \textbf{Preserve the Title}'' of such a section when you modify the Document means that it remains a section Entitled XYZ'' according to this definition. The Document may include Warranty Disclaimers next to the notice which states that this License applies to the Document. These Warranty Disclaimers are considered to be included by reference in this License, but only as regards disclaiming warranties: any other implication that these Warranty Disclaimers may have is void and has no effect on the meaning of this License. \begin{center} {\Large\bf 2. VERBATIM COPYING\par} %% \phantomsection %% \addcontentsline{toc}{section}{2. VERBATIM COPYING} \end{center} You may copy and distribute the Document in any medium, either commercially or noncommercially, provided that this License, the copyright notices, and the license notice saying this License applies to the Document are reproduced in all copies, and that you add no other conditions whatsoever to those of this License. You may not use technical measures to obstruct or control the reading or further copying of the copies you make or distribute. However, you may accept compensation in exchange for copies. If you distribute a large enough number of copies you must also follow the conditions in section~3. You may also lend copies, under the same conditions stated above, and you may publicly display copies. \begin{center} {\Large\bf 3. COPYING IN QUANTITY\par} %% \phantomsection %% \addcontentsline{toc}{section}{3. COPYING IN QUANTITY} \end{center} If you publish printed copies (or copies in media that commonly have printed covers) of the Document, numbering more than 100, and the Document's license notice requires Cover Texts, you must enclose the copies in covers that carry, clearly and legibly, all these Cover Texts: Front-Cover Texts on the front cover, and Back-Cover Texts on the back cover. Both covers must also clearly and legibly identify you as the publisher of these copies. The front cover must present the full title with all words of the title equally prominent and visible. You may add other material on the covers in addition. Copying with changes limited to the covers, as long as they preserve the title of the Document and satisfy these conditions, can be treated as verbatim copying in other respects. If the required texts for either cover are too voluminous to fit legibly, you should put the first ones listed (as many as fit reasonably) on the actual cover, and continue the rest onto adjacent pages. If you publish or distribute Opaque copies of the Document numbering more than 100, you must either include a machine-readable Transparent copy along with each Opaque copy, or state in or with each Opaque copy a computer-network location from which the general network-using public has access to download using public-standard network protocols a complete Transparent copy of the Document, free of added material. If you use the latter option, you must take reasonably prudent steps, when you begin distribution of Opaque copies in quantity, to ensure that this Transparent copy will remain thus accessible at the stated location until at least one year after the last time you distribute an Opaque copy (directly or through your agents or retailers) of that edition to the public. It is requested, but not required, that you contact the authors of the Document well before redistributing any large number of copies, to give them a chance to provide you with an updated version of the Document. \begin{center} {\Large\bf 4. MODIFICATIONS\par} %% \phantomsection %% \addcontentsline{toc}{section}{4. MODIFICATIONS} \end{center} You may copy and distribute a Modified Version of the Document under the conditions of sections 2 and 3 above, provided that you release the Modified Version under precisely this License, with the Modified Version filling the role of the Document, thus licensing distribution and modification of the Modified Version to whoever possesses a copy of it. In addition, you must do these things in the Modified Version: \begin{itemize} \item[A.] Use in the Title Page (and on the covers, if any) a title distinct from that of the Document, and from those of previous versions (which should, if there were any, be listed in the History section of the Document). You may use the same title as a previous version if the original publisher of that version gives permission. \item[B.] List on the Title Page, as authors, one or more persons or entities responsible for authorship of the modifications in the Modified Version, together with at least five of the principal authors of the Document (all of its principal authors, if it has fewer than five), unless they release you from this requirement. \item[C.] State on the Title page the name of the publisher of the Modified Version, as the publisher. \item[D.] Preserve all the copyright notices of the Document. \item[E.] Add an appropriate copyright notice for your modifications adjacent to the other copyright notices. \item[F.] Include, immediately after the copyright notices, a license notice giving the public permission to use the Modified Version under the terms of this License, in the form shown in the Addendum below. \item[G.] Preserve in that license notice the full lists of Invariant Sections and required Cover Texts given in the Document's license notice. \item[H.] Include an unaltered copy of this License. \item[I.] Preserve the section Entitled History'', Preserve its Title, and add to it an item stating at least the title, year, new authors, and publisher of the Modified Version as given on the Title Page. If there is no section Entitled History'' in the Document, create one stating the title, year, authors, and publisher of the Document as given on its Title Page, then add an item describing the Modified Version as stated in the previous sentence. \item[J.] Preserve the network location, if any, given in the Document for public access to a Transparent copy of the Document, and likewise the network locations given in the Document for previous versions it was based on. These may be placed in the History'' section. You may omit a network location for a work that was published at least four years before the Document itself, or if the original publisher of the version it refers to gives permission. \item[K.] For any section Entitled Acknowledgements'' or Dedications'', Preserve the Title of the section, and preserve in the section all the substance and tone of each of the contributor acknowledgements and/or dedications given therein. \item[L.] Preserve all the Invariant Sections of the Document, unaltered in their text and in their titles. Section numbers or the equivalent are not considered part of the section titles. \item[M.] Delete any section Entitled Endorsements''. Such a section may not be included in the Modified Version. \item[N.] Do not retitle any existing section to be Entitled Endorsements'' or to conflict in title with any Invariant Section. \item[O.] Preserve any Warranty Disclaimers. \end{itemize} If the Modified Version includes new front-matter sections or appendices that qualify as Secondary Sections and contain no material copied from the Document, you may at your option designate some or all of these sections as invariant. To do this, add their titles to the list of Invariant Sections in the Modified Version's license notice. These titles must be distinct from any other section titles. You may add a section Entitled Endorsements'', provided it contains nothing but endorsements of your Modified Version by various parties---for example, statements of peer review or that the text has been approved by an organization as the authoritative definition of a standard. You may add a passage of up to five words as a Front-Cover Text, and a passage of up to 25 words as a Back-Cover Text, to the end of the list of Cover Texts in the Modified Version. Only one passage of Front-Cover Text and one of Back-Cover Text may be added by (or through arrangements made by) any one entity. If the Document already includes a cover text for the same cover, previously added by you or by arrangement made by the same entity you are acting on behalf of, you may not add another; but you may replace the old one, on explicit permission from the previous publisher that added the old one. The author(s) and publisher(s) of the Document do not by this License give permission to use their names for publicity for or to assert or imply endorsement of any Modified Version. \begin{center} {\Large\bf 5. COMBINING DOCUMENTS\par} %% \phantomsection %% \addcontentsline{toc}{section}{5. COMBINING DOCUMENTS} \end{center} You may combine the Document with other documents released under this License, under the terms defined in section~4 above for modified versions, provided that you include in the combination all of the Invariant Sections of all of the original documents, unmodified, and list them all as Invariant Sections of your combined work in its license notice, and that you preserve all their Warranty Disclaimers. The combined work need only contain one copy of this License, and multiple identical Invariant Sections may be replaced with a single copy. If there are multiple Invariant Sections with the same name but different contents, make the title of each such section unique by adding at the end of it, in parentheses, the name of the original author or publisher of that section if known, or else a unique number. Make the same adjustment to the section titles in the list of Invariant Sections in the license notice of the combined work. In the combination, you must combine any sections Entitled History'' in the various original documents, forming one section Entitled History''; likewise combine any sections Entitled Acknowledgements'', and any sections Entitled Dedications''. You must delete all sections Entitled Endorsements''. \begin{center} {\Large\bf 6. COLLECTIONS OF DOCUMENTS\par} %% \phantomsection %% \addcontentsline{toc}{section}{6. COLLECTIONS OF DOCUMENTS} \end{center} You may make a collection consisting of the Document and other documents released under this License, and replace the individual copies of this License in the various documents with a single copy that is included in the collection, provided that you follow the rules of this License for verbatim copying of each of the documents in all other respects. You may extract a single document from such a collection, and distribute it individually under this License, provided you insert a copy of this License into the extracted document, and follow this License in all other respects regarding verbatim copying of that document. \begin{center} {\Large\bf 7. AGGREGATION WITH INDEPENDENT WORKS\par} %% \phantomsection %% \addcontentsline{toc}{section}{7. AGGREGATION WITH INDEPENDENT WORKS} \end{center} A compilation of the Document or its derivatives with other separate and independent documents or works, in or on a volume of a storage or distribution medium, is called an aggregate'' if the copyright resulting from the compilation is not used to limit the legal rights of the compilation's users beyond what the individual works permit. When the Document is included in an aggregate, this License does not apply to the other works in the aggregate which are not themselves derivative works of the Document. If the Cover Text requirement of section~3 is applicable to these copies of the Document, then if the Document is less than one half of the entire aggregate, the Document's Cover Texts may be placed on covers that bracket the Document within the aggregate, or the electronic equivalent of covers if the Document is in electronic form. Otherwise they must appear on printed covers that bracket the whole aggregate. \begin{center} {\Large\bf 8. TRANSLATION\par} %% \phantomsection %% \addcontentsline{toc}{section}{8. TRANSLATION} \end{center} Translation is considered a kind of modification, so you may distribute translations of the Document under the terms of section~4. Replacing Invariant Sections with translations requires special permission from their copyright holders, but you may include translations of some or all Invariant Sections in addition to the original versions of these Invariant Sections. You may include a translation of this License, and all the license notices in the Document, and any Warranty Disclaimers, provided that you also include the original English version of this License and the original versions of those notices and disclaimers. In case of a disagreement between the translation and the original version of this License or a notice or disclaimer, the original version will prevail. If a section in the Document is Entitled Acknowledgements'', Dedications'', or History'', the requirement (section~4) to Preserve its Title (section~1) will typically require changing the actual title. \begin{center} {\Large\bf 9. TERMINATION\par} %% \phantomsection %% \addcontentsline{toc}{section}{9. TERMINATION} \end{center} You may not copy, modify, sublicense, or distribute the Document except as expressly provided under this License. Any attempt otherwise to copy, modify, sublicense, or distribute it is void, and will automatically terminate your rights under this License. However, if you cease all violation of this License, then your license from a particular copyright holder is reinstated (a) provisionally, unless and until the copyright holder explicitly and finally terminates your license, and (b) permanently, if the copyright holder fails to notify you of the violation by some reasonable means prior to 60 days after the cessation. Moreover, your license from a particular copyright holder is reinstated permanently if the copyright holder notifies you of the violation by some reasonable means, this is the first time you have received notice of violation of this License (for any work) from that copyright holder, and you cure the violation prior to 30 days after your receipt of the notice. Termination of your rights under this section does not terminate the licenses of parties who have received copies or rights from you under this License. If your rights have been terminated and not permanently reinstated, receipt of a copy of some or all of the same material does not give you any rights to use it. \begin{center} {\Large\bf 10. FUTURE REVISIONS OF THIS LICENSE\par} %% \phantomsection %% \addcontentsline{toc}{section}{10. FUTURE REVISIONS OF THIS LICENSE} \end{center} The Free Software Foundation may publish new, revised versions of the GNU Free Documentation License from time to time. Such new versions will be similar in spirit to the present version, but may differ in detail to address new problems or concerns. See \texttt{http://www.gnu.org/copyleft/}. Each version of the License is given a distinguishing version number. If the Document specifies that a particular numbered version of this License or any later version'' applies to it, you have the option of following the terms and conditions either of that specified version or of any later version that has been published (not as a draft) by the Free Software Foundation. If the Document does not specify a version number of this License, you may choose any version ever published (not as a draft) by the Free Software Foundation. If the Document specifies that a proxy can decide which future versions of this License can be used, that proxy's public statement of acceptance of a version permanently authorizes you to choose that version for the Document. \begin{center} {\Large\bf 11. RELICENSING\par} %% \phantomsection %% \addcontentsline{toc}{section}{11. RELICENSING} \end{center} Massive Multiauthor Collaboration Site'' (or MMC Site'') means any World Wide Web server that publishes copyrightable works and also provides prominent facilities for anybody to edit those works. A public wiki that anybody can edit is an example of such a server. A Massive Multiauthor Collaboration'' (or MMC'') contained in the site means any set of copyrightable works thus published on the MMC site. CC-BY-SA'' means the Creative Commons Attribution-Share Alike 3.0 license published by Creative Commons Corporation, a not-for-profit corporation with a principal place of business in San Francisco, California, as well as future copyleft versions of that license published by that same organization. Incorporate'' means to publish or republish a Document, in whole or in part, as part of another Document. An MMC is eligible for relicensing'' if it is licensed under this License, and if all works that were first published under this License somewhere other than this MMC, and subsequently incorporated in whole or in part into the MMC, (1) had no cover texts or invariant sections, and (2) were thus incorporated prior to November 1, 2008. The operator of an MMC Site may republish an MMC contained in the site under CC-BY-SA on the same site at any time before August 1, 2009, provided the MMC is eligible for relicensing. \begin{center} {\Large\bf ADDENDUM: How to use this License for your documents\par} %% \phantomsection %% \addcontentsline{toc}{section}{ADDENDUM: How to use this License for your documents} \end{center} To use this License in a document you have written, include a copy of the License in the document and put the following copyright and license notices just after the title page: \bigskip \begin{quote} Copyright \copyright{} YEAR YOUR NAME. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled GNU Free Documentation License''. \end{quote} \bigskip If you have Invariant Sections, Front-Cover Texts and Back-Cover Texts, replace the with \dots\ Texts.''\ line with this: \bigskip \begin{quote} with the Invariant Sections being LIST THEIR TITLES, with the Front-Cover Texts being LIST, and with the Back-Cover Texts being LIST. \end{quote} \bigskip If you have Invariant Sections without Cover Texts, or some other combination of the three, merge those two alternatives to suit the situation. If your document contains nontrivial examples of program code, we recommend releasing these examples in parallel under your choice of free software license, such as the GNU General Public License, to permit their use in free software.]]>