# Linear Algebra via Exterior Products: book web site

This is the official web site for the free textbook "**Linear Algebra via Exterior Products**" (2010) by Sergei Winitzki.

### Summary

This 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 book

This 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. This *invariant* 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 project

This 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 book

As 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.

### Errata

In 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.)
- In Sec. 1.5.1, first paragraph, in the phrase "It is easy to check that this is a linear and one-to-one map of the subspace {(v,0) | v ∈ V } to V , and that Pˆ is a projector" the symbol Pˆ must be replaced by PˆV. Also, P is not a projector because it maps between different spaces. The phrase "that P is a projector" should be deleted. (Reported by Christophe Louargant <christo...>.)
- At the beginning of Sec. 1.6.1, unnumbered formula at top of page 39 has a mistake where u_i must be replaced by u_j in the last term. (Reported by Christophe Louargant <christo...>.)
- 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}, ...,**v**} and {_{m}**u**_{1}, ...,**u**} , whereas it should refer to {_{n}**v**_{1}, ...,**v**} and {_{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.)

### What the readers say

* http://mathoverflow.net/questions/17521/any-reference-on-multilinear-algebra*

Dear mingming, here are three excellent books.

...

3) Finally there is an amazingly original free book by Sergei Winitzki , *Linear Algebra via Exterior Products*. Here is the link

I couldn't easily find 2), but 3) does indeed seem very nice. I still have fear of the exterior product, and this book seems like a great way to mitigate it. It's unfortunate it doesn't seem to get to integration of forms, though. – Ilya Grigoriev Mar 9 at 0:57

* http://forums.xkcd.com/viewtopic.php?f=17&t=56912&view=previous*

by **t0rajir0u** » Thu Feb 25, 2010 4:06 am UTC

If you want a really conceptual explanation as to what's going on, you probably want to think of the determinant as the unique **alternating multilinear functional** (up to a constant) on n vectors in n-space. Why should area / volume be alternating and multilinear? ... There's a lot of interesting stuff here, most of which an introductory course in linear algebra will never get to.

If you're really interested in learning more about this point of view, you might want to try Sergei Winitzki's Linear Algebra via Exterior Products. But this book will take some patience and mathematical maturity to understand.

...

If you read Winitzki you'll learn that the cross product and the determinant are both special cases of a more general construction.

* http://qchu.wordpress.com/2009/06/01/non-canonical-isomorphisms/*

June 1, 2009

This isomorphism is called canonical because, while it can be defined by an arbitrary choice of basis, the final isomorphism does not depend on the choice of basis.

...

I didn’t finally get around to learning all of this stuff until I read Linear Algebra via Exterior Products by Sergei Winitzki, which is freely available at the URL; you can find all the details of what I’ve been talking about there. I found it an extremely interesting supplement to what I already knew about linear algebra; in particular, it “explains” the determinant formula in a way that I haven’t seen a standard linear algebra text try to tackle (via, as the title suggests, the exterior product).

*http://math.stackexchange.com/questions/25588/wedge-product-a-novel-interpretation-or-just-plain-wrong*

answered Mar 7, 2011 at 20:39

5917

The best introduction I know of to the exterior product is Sergei Winitzki's *free* book Linear Algebra via Exterior Products. Chapter 2 in particular I think addresses all of your questions (it is unclear how much of Chapter 1 you need to read in order to read Chapter 2, I guess that depends on how much linear algebra you've had).

### Terms and conditions for distribution

This book is copyrighted by myself and simultaneously published under a **free license** (GNU FDL). This license is generous: you may download the PDF file attached to this page, print that file if you want, email it to your friends, or distribute it in any form - even print it and sell it yourself - *as long as you adhere to the license*. The license allows you to copy the book for free, to get the complete source code for the book, and even to make changes and distribute modified versions. In return,* you may not prohibit other people from copying the book* further (and making further modifications!). All copies of the book are subjected to the same license. In this way, the license guarantees that the book (and any modified versions) will always remain free for everyone to copy or to use in any way they see fit.

Since nowadays there are only very few books published under a free license such as the FDL, and since most readers are most probably not familiar with these issues, I would like to explain in some detail the motivation and the nature of this license.

The conditions of a free license differ dramatically from the conditions normally applied to printed books. Usually, readers are not allowed to copy books at all - the publisher has an *exclusive right *to print a book or to make any copies of it. In other words, the publisher has a *monopoly* on the distribution of the book. Accordingly, the price of a given book is not determined by the law of supply and demand - it is a monopoly price. The only way to get hold of a book is to buy it at the publisher's monopoly price (or buy the book used; however, the used book pricing is often influenced by the price of a new book).

Even more severe restrictions usually apply to electronic books. We need to keep in mind that most books today are* first *produced electronically, e.g., as PDF files - and only then printed on paper. Therefore, it is very easy for publishers to make new books available as PDF files or in another electronic format. Nevertheless, most science publishers sell PDF books at about the same high price as paper books. Moreover, if you buy a paper copy of a book, usually you are *not allowed* to have an electronic (say, PDF) copy of the same book unless you *buy* a PDF copy separately. You can, of course, pay twice and buy an electronic copy together with a paper copy, but the electronic copy comes with a non-free license: usually, you are not allowed to print the PDF file or copy text out of it!

Moreover, the e-book file - the file you think you *``bought" - *may have technologically implemented restrictions on it. This file may "expire" at a set time in the future, or it may be viewable only a certain number of times or only on certain computers or book readers. The e-book file viewer may report you to some monitoring service every time you open the file, or it may impose other arbitrary restrictions on the use of the book. (These restrictions are usually called "security features" although they do not actually provide any security.) In a number of years, the copyright on the book may expire but you still will not be able to use the e-book file you *bought *because the technologically implemented restrictions will not expire! (In the US, the DMCA makes it illegal for you to remove these technological restrictions, *regardless* of the copyright status of the file - even if you are yourself the author of the book!)

To make it more clear: these technologically implemented restrictions on the computer files are *not constrained* by the copyright law - or by any law whatsoever. On the contrary, *you* are constrained by the law not to circumvent any of these restrictions - even if you would otherwise have the right to do so, e.g. when the book copyright expires or if you are the author. Richard Stallman explained what will happen if books become non-free in this way.

So the difference between the copyright usually applied to books (with a restrictive license) and the copyright combined with a free license is the following:

- The usual copyright
*prohibits*you from copying the book yourself, from using an electronic version of the book (if you purchase it) as you see fit, from making changes to the book (e.g. reformatting), and from distributing any copies of the book*in any way*- even non-commercially or privately to family and friends. You are allowed to copy not more than 10% of the book contents ("fair use") privately for research and teaching purposes. - The free license
*allows*you to copy for free, to make changes, and to distribute the book (either original*or modified*versions) without restrictions in any form (paper or PDF). You are allowed to print and sell paper copies, or to give copies away for free. But you are not allowed to distribute the book under an incompatible license (e.g., under a non-free license) or claim that you are the original author, or stop further distribution of the book.

There exist several free licenses, such as the FDL or the various "Creative Commons" or "Open Access" licenses, or even a public domain license. Why did I choose the FDL? My reasons were:

- I would like to make sure that the text stays free forever. The FDL allows everyone to make changes, translations, compilations based on my text, and so on, and to distribute modified versions, but only under the same license. Other licenses do not allow this and/or do not sufficiently protect the future freedom of the text. (For public domain texts, malicious people could create a small revision and make the entire text effectively non-free. For CC texts, revisions and derivative works are not automatically allowed and/or are not guaranteed to be free.)
- The FDL
*requires*everyone to distribute the entire machine-readable source code of the text. Moreover, the source*must*be provided in a format suitable and preferable for human editing (rather than in some obfuscated format). Other free licenses do not require distribution of the complete source code in a form preferable for editing. A typical example when the source code is not properly distributed is found in some "Open Access" books converted from LaTeX but available only as HTML with embedded JPEG diagrams or equations. If you print this kind of book from your browser, it will consume a lot of paper, will look quite ugly, and will be difficult to read. Preparing a standard print-quality version of such a book would require editing of the entire text by hand and retyping the equations - a very time-consuming task. This effectively prevents re-use of the book.

The book "*Linear Algebra via Exterior Products*" is now being sold through lulu.com at a low price; lulu.com is an on-demand publisher who respects free licenses and specifically accepts the FDL. So if you buy a printed copy from them, you are doing it by choice, because you want to have a bound paper copy of the book. It will probably cost you more to print and bind it yourself or in a print shop than if you order through lulu.com. I get $1 for each printed copy sold by lulu.com, and I get nothing for downloaded copies.

In compliance with the free license, you may also download a PDF file of the book, as well as the full source code to the book. You may download the files from lulu.com, from this web page, or from another web site. You may then upload the PDF file to any other web site, or print and distribute paper copies, *without paying me a cent* - and without asking my permission: the permission is already granted. I provide both the LaTeX and LyX sources of the book, to facilitate further editing (the entire book was produced with LyX, so the preferred way to make changes is to edit the LyX file rather than the LaTeX file).

If you find an error in the book, you may correct it by modifying the LaTeX file yourself and distributing modified versions, and/or you may let me know about the error. If you report errors to me, you can then be sure that your effort will not be lost because all future revisions of the book will be again available under the same license. This is what a **free book** stands for: "free" as in "freedom."

### Table of contents at a glance

Preface

0 Introduction and summary

0.1 Notation

0.2 Sample quiz problems

0.3 A list of results

1 Linear algebra without coordinates

1.1 Vector spaces

1.2 Linear maps in vector spaces

1.3 Subspaces

1.4 Isomorphisms of vector spaces

1.5 Direct sum of vector spaces

1.6 Dual (conjugate) vector space

1.7 Tensor product of vector spaces

1.8 Linear maps and tensors

1.9 Index notation for tensors

1.10 Dirac notation for vectors and covectors

2 Exterior product

2.1 Motivation

2.2 Exterior product

2.3 Properties of the spaces \wedge ^k V

3 Basic applications

3.1 Determinants through permutations: the hard way

3.2 The space \wedge^N V and oriented volume

3.3 Determinants of operators

3.4 Determinants of square tables

3.5 Solving linear equations

3.6 Vandermonde matrix

3.7 Multilinear actions in exterior powers

3.8 Trace

3.9 Characteristic polynomial

4 Advanced applications

4.1 The space \wedge{N -1} V

4.2 Algebraic complement (adjoint) and beyond

4.3 Cayley-Hamilton theorem and beyond

4.4 Functions of operators

4.5 Formulas of Jacobi and Liouville

4.6 Jordan canonical form

4.7 * Construction of projectors onto Jordan cells

5 Scalar product

5.1 Vector spaces with scalar product

5.2 Orthogonal subspaces

5.3 Orthogonal transformations

5.4 Applications of exterior product

5.5 Scalar product in \wedge^k V

5.6 Scalar product for complex spaces

5.7 Antisymmetric operators

5.8 * Pfaffians

A Complex numbers

A.1 Basic definitions

A.2 Geometric representation

A.3 Analytic functions

A.4 Exponent and logarithm

B Permutations

C Matrices

C.1 Definitions

C.2 Matrix multiplication

C.3 Linear equations

C.4 Inverse matrix

C.5 Determinants

C.6 Tensor product

D Distribution of this text

D.1 Motivation

D.2 GNU Free Documentation License

Index