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

-<p>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.</p>

+<p>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.</p>

<p>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.</p>

<introduction>

<p>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.</p>

-<p>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.</p>

+<p>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.</p>

-<p>While there is an abundance of guidance in the use of the software system, <miscref type="sage" text="Sage" />, there is no attempt to address the problems of numerical linear algebra, which are basically 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.</p>

+<p>While there is an abundance of guidance in the use of the software system, <miscref type="sage" text="Sage" />, 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.</p>

</introduction>

<plainsubsection>

<title>How to Use This Book</title>

-<p>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 <em>before</em> class and answer the section's reading questions.</p>

+<p>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 <em>before</em> class and then answer the section's reading questions.</p>

-<p>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.</p>

+<p>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.</p>

-<p>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.</p>

+<p>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.</p>

-<p>Exercises come in three flavors, indicated by the first letter of their label. <q>C</q> indicates a problem that is essentially computational. <q>T</q> represents a problem that is more theoretical, usually requiring a solution that is as rigorous as a proof. <q>M</q> stands for problems that are <q>medium</q>, <q>moderate</q>, <q>midway</q>, <q>mediate</q> or <q>median</q>, but never <q>mediocre</q>. 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.</p>

+<p>Exercises come in three flavors, indicated by the first letter of their label. <q>C</q> indicates a problem that is essentially computational. <q>T</q> represents a problem that is more theoretical, usually requiring a solution that is as rigorous as a proof. <q>M</q> stands for problems that are <q>medium</q>, <q>moderate</q>, <q>midway</q>, <q>mediate</q> or <q>median</q>, but never <q>mediocre</q>. 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.</p>

-<p>The Archetypes are a collection of twenty-four archetypical examples. The open source lexical database, WordNet, defines an archetype as <q>something that serves as a model or a basis for making copies</q>. 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.</p>

+<p>The Archetypes are a collection of twenty-four archetypical examples. The open source lexical database, WordNet, defines an archetype as <q>something that serves as a model or a basis for making copies.</q> 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.</p>

</plainsubsection>

<plainsubsection>

<plainsubsection>

<title>Freedom</title>

-<p>This book is copyrighted by its author. Some would say it is his <q>intellectual property,</q> 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 <miscref type="license" text="GNU Free Documentation License (GFDL)" />. 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 <q>out-of-print.</q> 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.</p>

+<p>This book is copyrighted by its author. Some would say it is his <q>intellectual property,</q> 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 <miscref type="license" text="GNU Free Documentation License (GFDL)" />. 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 <q>out-of-print.</q> 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.</p>

<p>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.</p>

<plainsubsection>

<title>To the Instructor</title>

-<p>The first half of this text (through <acroref type="chapter" acro="M" />) is basically a course in matrix algebra, though the foundation of some more advanced ideas is also being formed in these early sections. Vectors are presented exclusively as column vectors (since we also have the typographic freedom to avoid writing a column vector inline as the transpose of a row vector), 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.</p>

+<p>The first half of this text (through <acroref type="chapter" acro="M" />) is basically a course in matrix algebra, though the foundation of some more advanced ideas is also being formed in these early sections (such as <acroref type="theorem" acro="NMUS" />, 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.</p>

-<p>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 (<acroref type="chapter" acro="VS" />). Once bases and dimension have been explored in the context of vector spaces, linear transformations and their matrix representation follow. The goal of the book is <acroref type="chapter" acro="R" />, which introduces the student to representations of vectors and matrices, along with orthonormal diagonalization (a Schur decomposition over an algebraically closed field). This final chapter pulls together all the imortant ideas of the previous chapter.</p>

+<p>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 (<acroref type="chapter" acro="VS" />). 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 <acroref type="chapter" acro="R" />, 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.</p>

<p>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 <acroref type="section" acro="CNO" /> provides a quick review.</p>

<p>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 (<miscref type="definition" text="Definitions" /> and <miscref type="theorem" text="Theorems" /> in the reference section of the book). Along the way, there are discussions of some more important ideas relating to formulating proofs (<miscref type="technique" text="Proof Techniques" />), which is partly advice and partly a primer on logic.</p>

-<p>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.</p>

+<p>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.</p>

-<p><miscref type="sage" text="Sage" /> 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 as 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.</p>

+<p><miscref type="sage" text="Sage" /> 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.</p>

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