\thispagestyle{plain} \setlength{\parskip}{.7ex} \bigskip \vspace*{1.25in} \noindent{\Huge\bf Preface} \vspace*{.4in} % \bigskip \par\noindent This book helps students to master the material of a standard US undergraduate linear algebra course. The material is standard in that the subjects covered are Gaussian reduction, vector spaces, linear maps, determinants, and eigenvalues and eigenvectors. Another standard is book's audience: sophomores or juniors, usually with a background of at least one semester of calculus. The help that it gives to students comes from taking a developmental approach\Dash this book's presentation emphasizes motivation and naturalness, using many examples as well as extensive and careful exercises. The developmental approach is what most recommends this book so I will elaborate. The courses at the beginning of a mathematics program focus less on theory and more on calculating. Later courses ask for mathematical maturity:~the ability to follow different types of arguments, a familiarity with the themes that underlie many mathematical investigations such as elementary set and function facts, and a capacity for some independent reading and thinking. Some programs have a separate course devoted to developing maturity and some do not. In either case, linear algebra is an ideal spot to work on this transition. It comes early in a program so that progress made here pays off later, but also comes late enough that students are serious about mathematics. The material is accessible, coherent, and elegant. There are a variety of argument styles, including direct proofs, proofs by contradiction, and proofs by induction. And, examples are plentiful. Helping readers start the transition to being serious students of mathematics requires taking the mathematics seriously, so all of the results here are proved. On the other hand, we cannot assume that students have already arrived and so in contrast with more abstract texts this book is filled with examples, often quite detailed. Some linear algebra books begin with extensive computations of linear systems, matrix multiplications, and determinants. Then, when vector spaces and linear maps finally appear and definitions and proofs start, the abrupt change can bring students to an abrupt stop. While this book starts with a computational topic, linear reduction, from the first we do more than compute. The linear reduction chapter includes the proofs needed to justify what we are computing. Then, with the linear systems work as motivation so that the study of linear combinations seems natural, the second chapter starts with the definition of a real vector space. In the schedule below, this occurs by the end of the third week. Another example of this book's emphasis on motivation and naturalness is that the third chapter on linear maps does not begin with the definition of homomorphism. Rather, we start with the definition of isomorphism, which is natural: students themselves observe that some spaces are just like'' others. After that, the next section takes the reasonable step of isolating the operation-preservation idea to define homomorphism. This approach loses mathematical slickness, but it is a good trade because it gives to students a large gain in sensibility. A student progresses most in mathematics while doing exercises so the ones here have gotten close attention. Problem sets start with simple checks and range up to reasonably involved proofs. Since instructors usually assign about a dozen exercises I have tried to put about two dozen in each set, thereby giving a selection. There are even a few that are puzzles taken from various journals, competitions, or problems collections. These are marked with a \puzzlemark' and as part of the fun the original wording has been retained as much as possible. That is, as with the rest of the book, the exercises are aimed to both build an ability at, and help students experience the pleasure of, \emph{doing} mathematics. Students should see how the ideas arise and should be able to picture themselves doing the same type of work. %\vspace*{.5in} \medskip \noindent{\bf Applications and computers.} %\smallskip The point of view taken here, that students should think of linear algebra as about vector spaces and linear maps, is not taken to the complete exclusion of others. Applications and computing are interesting and vital aspects of the subject. Consequently, each of this book's chapters closes with a few topics in those areas. They are brief enough that an instructor can do one in a day's class or can assign them as independent or small-group projects. Most simply give a reader a taste of the subject, discuss how linear algebra comes in, point to some further reading, and give a few exercises. Whether they figure formally in the course or not, these help readers see for themselves that linear algebra is a tool that a professional must have. \medskip \noindent{\bf The license.} This book is freely available. In particular, instructors can print copies for students and sell them at a college bookstore. See \url{http://joshua.smcvt.edu/linearalgebra} for the details. That page also contains this book's latest version, answers to the exercises, and the \LaTeX\ source. I am very glad for bug reports. I save them and periodically issue revisions. See the contact information on that page. \newcommand{\classday}[1]{\textsc{#1}} \newcommand{\colwidth}{1.25in} %\vspace*{.5in} \medskip \noindent{\bf If you are reading this book on your own.} %\smallskip % This book's emphasis on motivation and development, and its availibility, make it widely used for self-study. If you are an independent student then you may find some advice helpful. In particular, while an experienced instructor knows what subjects and pace suit their class, you may find useful these timetables for a semester. The first focuses on core material. \begin{center} \small \begin{tabular}{r|*{2}{p{\colwidth}}l} \textit{week} &\textit{Monday} &\textit{Wednesday} &\textit{Friday} \\ \hline 1 &One.I.1 &One.I.1, 2 &One.I.2, 3 \\ 2 &One.I.3 &One.II.1 &One.II.2 \\ 3 &One.III.1, 2 &One.III.2 &Two.I.1 \\ 4 &Two.I.2 &Two.II &Two.III.1 \\ 5 &Two.III.1, 2 &Two.III.2 &\classday{exam} \\ 6 &Two.III.2, 3 &Two.III.3 &Three.I.1 \\ 7 &Three.I.2 &Three.II.1 &Three.II.2 \\ 8 &Three.II.2 &Three.II.2 &Three.III.1 \\ 9 &Three.III.1 &Three.III.2 &Three.IV.1, 2 \\ 10 &Three.IV.2, 3, 4 &Three.IV.4 &\classday{exam} \\ 11 &Three.IV.4, Three.V.1 &Three.V.1, 2 &Four.I.1, 2 \\ 12 &Four.I.3 &Four.II &Four.II \\ 13 &Four.III.1 &Five.I &Five.II.1 \\ 14 &Five.II.2 &Five.II.3 &\classday{review} \end{tabular} \end{center} The second is more ambitious. It supposes that you know section One.II, the elements of vectors. \begin{center} \small \begin{tabular}{r|*{2}{p{\colwidth}}l} \textit{week} &\textit{Monday} &\textit{Wednesday} &\textit{Friday} \\ \hline 1 &One.I.1 &One.I.2 &One.I.3 \\ 2 &One.I.3 &One.III.1, 2 &One.III.2 \\ 3 &Two.I.1 &Two.I.2 &Two.II \\ 4 &Two.III.1 &Two.III.2 &Two.III.3 \\ 5 &Two.III.4 &Three.I.1 &\classday{exam} \\ 6 &Three.I.2 &Three.II.1 &Three.II.2 \\ 7 &Three.III.1 &Three.III.2 &Three.IV.1, 2 \\ 8 &Three.IV.2 &Three.IV.3 &Three.IV.4 \\ 9 &Three.V.1 &Three.V.2 &Three.VI.1 \\ 10 &Three.VI.2 &Four.I.1 &\classday{exam} \\ 11 &Four.I.2 &Four.I.3 &Four.I.4 \\ 12 &Four.II &Four.II, Four.III.1 &Four.III.2, 3 \\ 13 &Five.II.1, 2 &Five.II.3 &Five.III.1 \\ 14 &Five.III.2 &Five.IV.1, 2 &Five.IV.2 \end{tabular} \end{center} In the table of contents I have marked subsections as optional if some instructors will pass over them in favor of spending more time elsewhere. You might pick one or two topics that appeal to you from the end of each chapter. You'll get more from these if you have access to software for calculations. I recommend \textit{Sage}, freely available from \url{http://sagemath.org}. My main advice is: do many exercises. I have marked a good sample with \recommendationmark's in the margin. For all of them, you must justify your answer either with a computation or with a proof. Be aware that few inexperienced people can write correct proofs; try to find a knowledgable person to work with you on these. \bigskip Finally, a caution for all students, independent or not:~I cannot overemphasize how much the statement, which I sometimes hear, `I understand the material but it's only that I have trouble with the problems''\spacefactor=1000\ % reveals a misconception. Being able to do things with the ideas is their entire point. The quotes below express this sentiment admirably. They capture the essence of both the beauty and the power of mathematics and science in general, and of linear algebra in particular. (I took the liberty of formatting them as verse). \bigskip \par\noindent\begin{tabular}[t]{@{}l@{}} \textit{I know of no better tactic} \\ \textit{\ than the illustration of exciting principles} \\ \textit{by well-chosen particulars.} \\ \hspace*{1in}\textit{--Stephen Jay Gould} \end{tabular} \bigskip \par\noindent \begin{tabular}[t]{@{}l@{}} \textit{If you really wish to learn} \\ \textit{\ then you must mount the machine} \\ \textit{\ and become acquainted with its tricks} \\ \textit{by actual trial.} \\ \hspace*{1in}\textit{--Wilbur Wright} \end{tabular} \par\ \hfill\begin{tabular}[t]{@{}l@{}} Jim Hef{}feron \\ Mathematics, Saint Michael's College \\ Colchester, Vermont\ USA 05439 \\ \texttt{http://joshua.smcvt.edu} \\ 2011-Dec-18 \end{tabular} \vfill \par\noindent\textit{Author's Note.} Inventing a good exercise, one that enlightens as well as tests, is a creative act and hard work. The inventor deserves recognition. But texts have traditionally not given attributions for questions. I have changed that here where I was sure of the source. I would be glad to hear from anyone who can help me to correctly attribute others of the questions.