Linear Algebra via Exterior Products" (2010) by Sergei Winitzki.## SummaryThis book is a pedagogical introduction to the coordinate-free approach in basic finite-dimensional linear algebra. Throughout this book, extensive use is made of the exterior (anticommutative, “wedge”) product of vectors: . The coordinate-free formalism and the exterior product, while somewhat more abstract, provide a deeper understanding of the classical results in linear algebra. The reader should be already familiar with the elementary array-based formalism of vector and matrix calculations, in order to fully appreciate the approach based on exterior products. The standard properties of determinants, the Pythagorean theorem for multidimensional volumes, the Liouville formula, the Hamilton-Cayley theorem, Pfaffians, as well as some results concerning eigenspace projectors are derived using exterior products, without cumbersome matrix calculations. For the benefit of students, every result is logically motivated and discussed. Exercises with some hints are provided.## Get the bookThis book is now available in printed form, published by lulu.com as a low-priced paperback or as a free PDF download. (The price of a printed copy is essentially the cost of printing and binding.) You can also order it at amazon.com if you prefer. (There was a Kindle edition, but the quality of Kindle rendering was inadequate for a text with equations, and Amazon withdrew it after a while.)The following files are available for free download and distribution (see attachments at the bottom of this page). All PDF files contain identical text except for the formatting. - linalg-ebook.pdf --- for on-screen reading. This file is formatted as a normal book (299 pages) and has hyperlinks and bookmarks for navigation.
- linalg-book-print.pdf --- for printing, formatted similarly to linalg-ebook.pdf but without hyperlinks/bookmarks and with somewhat larger page size. This file is large because it also contains a hyperlink-enabled PDF file as attachment. This is the PDF file (287 pages) you can download from lulu.com.
- Linear_algebra_exterior.pdf --- with hyperlinks/bookmarks, with embedded source, formatted in two columns to save paper (this file has only 127 pages).
Each of these files contains the source code for the entire book (linalg.tar.gz). See below for terms and conditions of distribution. (It's "free" as in "freedom".)
The LyX and LaTeX source code for the book is embedded as an attachment named linalg.tar.gz within the PDF files. If you would like to extract the attached source code from the PDF file but do not wish to use the Acrobat Reader program, you can extract this attachment with the free utility pdftk using a command such as pdftk Linear_algebra_exterior.pdf unpack_files
Then you can unpack the archive and build the book using the provided Makefile. For example: tar zxf linalg.tar.gz
make linalg-ebook.pdf ## From the preface"In a first course of linear algebra, one learns the various uses of matrices, for instance the properties of determinants, eigenvectors and eigenvalues, and methods for solving linear equations. The required calculations are straightforward (because, conceptually, vectors and matrices are merely "arrays of numbers") if cumbersome. However, there is a more abstract and more powerful approach: Vectors are elements of abstract vector spaces, and matrices represent linear transformations of vectors. Thisinvariant or coordinate-free approach is important in algebra and has found many applications in science. "The purpose of this book is to help the reader make a transition to the abstract coordinate-free approach, and also to give a hands-on introduction to exterior products, a powerful tool of linear algebra. I show how the coordinate-free approach together with exterior products can be used to clarify and extend the basic results of matrix algebra, at the same time avoiding all the laborious matrix calculations. "Here is a simple theorem that illustrates the advantages of the exterior product approach. Take a triangle oriented arbitrarily in three-dimensional space; the three orthogonal projections of this triangle are triangles in the three coordinate planes. Let S be the area of the initial triangle, and let A,B,C be the areas of the three projections. Then "If one uses bivectors to represent the oriented areas of the triangle and of its three projections, the statement is equivalent to the Pythagoras theorem in the space of bivectors, and the proof requires only a few straightforward definitions and checks. A generalization of this result to volumes of k-dimensional bodies embedded in N-dimensional spaces is then obtained with no extra work. I hope the readers will appreciate the beauty of an approach to linear algebra that allows us to obtain such results quickly and almost without calculations."## History of the projectThis book is an outgrowth of my explorations in basic linear algebra that I started back in 1991 while still an undergraduate student. I rediscovered determinants in the language of exterior product of vectors; it made much more sense than the matrix formulation using permutations. Eventually I was able to prove the Hamilton-Cayley theorem by using exterior product techniques. In 1992 or 1993 I wrote a short paper containing this proof and a few derivations of other standard results; this was my very first text written using TeX --- because LaTeX was not yet mature or widely known. That paper (the file lalg.ps.gz attached below) was never published because I felt that it is not useful to publish a new proof of one or two standard theorems.The next iteration happened around 2000 or so, when I produced a longer text with more details and proofs. I decided to start with a definition of determinants through the exterior product and try to follow the natural development of this approach. Indeed, it was possible to obtain most standard results of linear algebra without referring to any matrices or components of vectors, and without using the tensor index notation. Finally, I found a formula for a projector onto a Jordan cell knowing only the characteristic polynomial and one eigenvalue of the operator. (The projector is a polynomial in the operator with specially chosen coefficients. My formula did not always work, as I found much later, but the approach was right, and the mistake could be corrected.) A friend of mine, who is a professor of mathematics, got interested and encouraged me to put together my notes for the benefit of students. I wrote an extensive first chapter with definitions of vector spaces and tensor products, and beefed up the text with exercises. He started distributing my unfinished text to students as a course supplement. I have been working on the book on and off, in my free time. It was only in 2009 that I got around to finishing the text and converting it into a well-rounded textbook. The book now has a thorough discussion of the geometric interpretation of the exterior product in any dimensions, several new constructions (for instance, I derived the Jordan canonical form of an operator using exterior product techniques), and many new exercises. In May 2009 the book has been upgraded to version 0.9.6 and announced at several internet collections of free books. I got some feedback from a few readers (although I expected none, really). From version 0.9.6 to 1.1.4, the text has been thoroughly proofread and slightly expanded; between versions 1.1.4 and 1.2, the text was proofread and revised two more times in order to correct errors, while very little new material was added. A lot of effort was spent improving the text, changing small details, and making sure that the material is presented clearly and in a logically compelling sequence --- especially for people who are not already familiar with the chosen approach to linear algebra. I call this book a " textbook" because it is a self-contained, pedagogical presentation of material that is well known, rather than a report on new research. (Although this book uses an approach to linear algebra that is currently not widely adopted, this approach has been known for a very
long time and cannot be regarded as original.) One particularity of a textbook is that it usually does not have a proper bibliography, i.e. a collection of references that are crucial for understanding the presentation. A bibliography would be required if the readers had to consult other publications in order to find some relevant details or to learn about previous results; but in a textbook, no important details should be omitted, and the material should be explained from the ground up. I do suggest a couple of other textbooks on linear algebra, and infrequently refer to certain other publications in the footnotes. Nevertheless, these references are strictly optional and may be skipped.## Current status of the bookAs of version 1.2 (released on 4 January 2010), the book is finished. Typos and errors will be corrected if found or reported. This is the final version published under ISBN 978-1-4092-9496-2 and announced through Amazon and perhaps other online resellers. This version is also available in print and for download through lulu.com.## ErrataIn most books, there is room for improvement and errors (straightforward or subtle) to be corrected. Errors in scientific texts are nowadays frequently called "misprints." I am against this practice because it is dishonest. A true misprint would be "hte" instead of "the". This kind of error is so insignificant in scientific texts that one does not mention such errors at all. What is called a "misprint" in many contemporary texts is actually a minor error, such as a missing minus sign or factor 2, which is not especially important for the result being derived (and therefore goes unnoticed by the authors). Calling an error a "misprint" downplays the importance of error-free presentation, which is crucial in a book intended for beginners. Such errors need to be found and corrected because they will unnecessarily confuse many readers.Here I will collect the list of known errors in the book "Linear Algebra via Exterior Products." Please email me if you find other errors. I am grateful to all readers who notified me of errors in this book. - In Sec. 1.1.2, the definition of a "number field" in terms of axioms should explicitly mention the associativity of the operations + and *. (Reported by Jiri Matousek <matousek@...>.)
- In Sec. 1.1.2 (page 13), there is an incorrect property "a*1=1" listed (instead of "a*1=a"). (Reported by Joseph Ferrara <joe@te...>.)
- In Sec. 1.1.2 (page 15), in Definition V2, item 3, the associativity property should be added: λ*(μ*
**v**)=(λ*μ)***v**. (Reported by Dmitri Pavlov <pavlov239@...>.) - In Sec. 1.4, the definition of "canonically defined linear map" is not really a description of a property of a linear map. One cannot say precisely what it means that a map is "defined independently of a choice of basis". Rather, it is a property of the way we construct the map; it is useful to know that certain maps are constructed independently of a basis. (Prof. Matousek recommends that this "definition" be considered as heuristic at this point.)
- At the beginning of Sec. 1.6.1 there is a Remark to the effect that "the dual space V* may be larger or smaller than V" if V is infinite-dimensional. This remark is imprecise and should be revised: there are too many subtleties of functional analysis to talk about "larger" or "smaller" dual infinite-dimensional spaces at this point. It is safe to say that the dual space may be "larger" in the sense that a linear map V -> V* exists such that it is an injection but not a surjection. (Reported by Jiri Matousek.)
- In Sec. 1.7.3, in the proof of Lemma 3, after Eq. (1.23) the text "Then we apply the map f*... defined in Lemma 1" must be replaced by "... defined in Lemma 2." (Reported by Pablo Dominguez <pablodgz@...>.)
- In Sec. 1.7.3, the statement of the same Lemma 3 has an error: it refers to the sets {
, ...,**v**_{1}} and {**v**_{m}, ...,**u**_{1}} , whereas it should refer to {**u**_{n}, ...,**v**_{1}} and {**v**_{m}, ...,**w**_{1}**w**}. The proof is unaffected. (Reported by Andrew J. Ho <ho.andrew.j@...> and corrected by Yuxi Liu <yliu...@...>.)_{n} - In Sec. 2.3.4, the calculations in Example 1 and Example 2 contain errors in arithmetic. (Reported by Gabriel Aguirre <gabriel.agirre@...>.) Corrected calculations are as follows: In Example 1, the vector
**c**_{1}=**c**-2**b**is equal to -**e**_{2}+29**e**_{3}, rather than to -**e**_{2}+9**e**_{3}as printed. Hence, the vector**a**_{1}+6**c**_{1}equals 280**e**_{3}rather than 160**e**_{3}as printed. The final result has the coefficient -280 instead of -160. In Example 2, the vector**c**_{1}is replaced by**c**_{1}+3**b**_{1}, but this is incorrect because this does not eliminate the vector**e**_{2}. The correct elimination procedure must replace**b**_{1}by**b**_{1}+3**c**_{1}. The correct final answer for Example 2 is 2**e**_{1}**e**_{2}e_{3}-7**e**_{1}**e**_{2}e_{4}+3**e**_{1}**e**_{3}e_{4}+**e**_{2}e_{3}e_{4}. - In Sec. 4.3 I discuss the generalization of the Cayley-Hamilton theorem and make a statement to the effect that an identically vanishing polynomial in some operator can be found (this is the last statement in Theorem 2). This statement should be regarded as a conjecture rather than a known fact. I do prove that there is a particular system of algebraic equations satisfied by
*several*operators. However, I did not prove that this system of equations is reducible to a polynomial equation in a*single*operator, i.e. that one can eliminate all other variables from the system. This part of statement (actually, the entire Theorem 2) is not used anywhere in the text; in any case, I don't know how to prove this. Other parts of Theorem 2 are correct. (Noticed by myself.) - In Sec. 4.5.3, the first equation should read "exp G(x) = ..." instead of "G(x) = ...". (Noticed by myself.)
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