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The Dark Art of Linear Algebra

Seth Braver
Vector Vectorum Books
Publication Date: 
Number of Pages: 
[Reviewed by
Mark Hunacek
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I accepted the invitation to review this book in no small measure because of the title, which evoked, to me, images of Harry Potter. I suppose one should expect unusual titles in a book by Braver, who, after all, wrote another one titled Full Frontal Calculus. In any event, I was curious to see just what the title of this book meant. Having read the book, I’m still not exactly sure, but I don’t mind, because reading the book was an enjoyable experience. Braver’s writing style is very conversational and humorous, and one that I think will appeal to students. 
As an example, we need go no further than the Index, which contains references like “Mary’s Little \( \lambda \)” (and which directs you to a page on which is an exercise, discussing shears, and which asks you to discuss the effect of a shear on a picture of a sheep) and “How to Keep A Fool Busy” (which directs you to the very page on which this reference appears). 
There is also, of course, the text itself, which contains all sorts of humorous references, some of them likely to be understood only by professionals. A footnote in the Preface reads “Forgive me, great-great-grandfather Halmos. My textbooks have illustrations… Forgive me, grandfather Axler, for I have used determinants.” 
But as enjoyable as the book is to read, it is also quite idiosyncratic, and an instructor thinking of using this book as a text should be aware of its unusual features. For one thing, the approach to the subject does not follow that of a typical linear algebra book. Braver correctly points out that most linear algebra books follow one of two paths: one that starts with the definition of an abstract vector space (which the author describes as mathematically elegant but “opaque to beginners”) and a computational approach via systems of linear equations (“so mind-numbingly dull as to constitute a crime against art”). 
So, the author has developed a third approach, one that stresses the geometric intuition behind the basic ideas and which does not discuss linear equations until that underlying intuition has been mastered. The book begins with a chapter on vectors (objects in Euclidean space, thought of first as arrows and then, after the introduction of coordinates, n-tuples), and discusses (for these concepts) the notions of subspaces, linear dependence, linear independence, and bases. Then linear transformations and their matrix representations are introduced (again, geometrically; more on this later). This is followed by a chapter on determinants (again, defined geometrically, in terms of volumes). The next chapter discusses change of basis and linear isometries, after which there is a chapter titled “Eigenstuff”. The book ends with a chapter on orthogonal projection and least squares.
These topics are all pretty standard, but the method of presentation is most definitely not. Perhaps the most unusual feature of the book is the fact that abstract vector spaces, though briefly alluded to (in a two-page section titled “Abstract Linear Algebra: A Trailer”), are never formally defined in the text; the focus here is on the Euclidean spaces Rn and their subspaces. (Strang’s Linear Algebra and Its Applications lists the axioms for a vector space in the exercises, but at least they’re listed somewhere.)  It follows, of course, that the dot product takes center stage and replaces general inner products.
Other topics are also treated in an unusual way. A linear transformation, for example, is defined geometrically as a “grid transformation induced by dragging the tips of the standard basis vectors to new places.” It is mentioned that linear transformations “preserve linear combinations”, but the familiar equations \( T(v + w) = T(v) + T(w) \) and \( T(cv) = cT(v) \) are mentioned only in the aforementioned “Trailer”.  
This book also eschews the traditional theorem/proof format. In fact, both of these words are used sparingly in the text, and replaced instead by discussions, sometimes very intuitive. Exercises seldom call for proofs but do ask the student to do computations and offer explanations for why certain statements are true. The statement that a linear transformation fixes the origin, for example, which usually appears as one of the first theorems proved after linear transformations are defined, is not explicitly mentioned in the text at all, but does appear as a true-false exercise. (Selected answers to the exercises, by the way, appear in a 13-page appendix at the end of the text.) 
Whether this book should be adopted as a text for an undergraduate course depends entirely, I think, on the way the course is structured.  Any department that views the introductory linear algebra course as an introduction to proof would likely find this text completely unsuitable, but a university that offers two consecutive linear algebra courses (a very informal look at the subject that is followed by a more sophisticated approach) might find this book well worth looking at. Certainly, this book should not be used in colleges that offer only one introductory linear algebra course; it goes without saying that no math major should graduate from college without knowing what an abstract vector space is. Students who are taking, or planning on taking, a more traditional course in linear algebra, might well enjoy looking at this book as a source of geometric insight and intuition, and for that matter instructors of a more traditional course might also enjoy looking at it for tips on how to impart some of that intuition in their lectures. 
Mark Hunacek ( is a Teaching Professor Emeritus at Iowa State University. 
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