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 mase 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.

+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.

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.

@@ -225,22 +225,22 @@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 exercises, the book is ideal for self-study, for those of any age.

+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,

While there is an abundance of guidance in the use of the software system,

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 answer the section's reading questions.

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.

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. The PDF and print versions of the book contain similar cross-references but their utility is more limited.

+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. The PDF and print versions of the book contain fewer cross-references.

-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. Each chapter has a list of ten or so important items from the chapter, and you will find yourself recognizing some of these acronyms with no extra effort beyond the normal amount of study.

+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. 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. Resist the urge to peek early, which is why solutions are only available in the web version by clicking on a knowl. Working the exercises diligently is the best way to master the material.

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 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 guide you in this quest, though these are not comprehensive.

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 guide you in this quest, though these are not comprehensive.

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 monoply that is copyright, the author has granted you a license, the 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.

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 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.

@@ -266,17 +266,17 @@The first half of this text (through

The first half of this text (through

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 (

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 (

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

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 (

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 50 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.

+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.

-Tacoma, Washington

-August 2012