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Saturday, May 12, 2007

JoaLDG: Artin on matrices

Preface. This entry is a first part of two planned about the mathematician Emil Artin. It's a little heavy on the math, (no surprise!), so let's heed in advance some advice we're about to quote and remember to pass gently over the oppressive parts without letting ourselves be burdened by their gravity. Listen to the music and not the song -- or was it the other way around? -- never mind, they're both light and airy.


Emil Artin (1892-1968) was one of our most formidable expositors of mathematics for mathematicians. To give just a most obvious and striking example of this talent, there is a reason why all introductory texts on Galois theory sound the same, and that reason is that they all borrow very, very heavily from Artin's book on the same. Artin on this subject was original: it was he who reformulated the work of Evariste Galois (1811-1832) from a theory of the symmetries of roots of polynomials into a theory of the symmetries of field extensions. Considering how this view now completely dominates it is a little surprising to learn that it was only so recently developed – Artin's book Galois theory was published in 1942, from his lecture notes, being fruit of work from the preceding years.

(But maybe not too strange. A long parenthetical digression giving context could be placed here. Let it suffice to say that even the notion of a quotient group was only formalised in the 1920's, and one can hardly state the “fundamental theorem of Galois theory” as we know it today without understanding group-theoretically what a normal subgroup is.)

I don't know what I'm doing in the fall but there's a certain chance I'll be teaching a section of this linear algebra class I've been grading – and I surely will be doing so sometime before I graduate – so some things Artin has written, and one passage in particular which I'll quote presently, have been a little on my mind. How do you tell people about linear algebra? At heart all I can answer is: Really, the same way as you do for anything else. Karl Jaspers thought that the problem of communication was one of the fundamental problems of philosophy. But we needn't feel abstractly pessimistic or overburdened: there are plenty of fundamental problems we manage willy-nilly to cope with every day. We have twenty thousand purely practical facts to draw on. And in this case, one of them is Artin's legacy.

Artin's book Geometric algebra is curiously organised: he deposits in the first chapter, prior to the main subjects of the book, all the external tools and apparatuses he'll need in the sequel. ("Curiously?" Well, normal people would put this in an appendix.) In the very thoughtful short preface labelled “Suggestions for the use of this book,” he explains:
The most important point to keep in mind is the fact that Chapter I should be used mainly as a reference chapter for the proofs of certain isolated algebraic theorems. These proofs have been collected so as not to interrupt the main line of thought in later chapters.
He goes on to say that “the inexperienced reader should start right away with Chapter II,” which to me reads like an agreement that Chapter I ought to be adjacent to the other cover. (Is he saying that the experienced reader shouldn't start right away with Chapter II?) He continues on, to give some of the best advice possible for reading mathematics, namely,
This skipping [of “a few harder algebraic theorems” in “a first reading”] is another important point. It should be done whenever a proof seems too hard or whenever a theorem or a whole paragraph does not appeal [!] to the reader. In most cases he will be able to go on and later on he may return to the parts which were skipped.*
Probably the students will object: there is hardly time for all this, to first skip and then to come back. Perhaps so. It is certainly unrealistic to think that the students will understand something that's unlike anything they've ever seen before in time to give clear and concise solutions on the weekly problem sets. But by the end of the class there should no longer be any mystery about the material of the first week, and the gap between the end of lectures and the beginning of the exam period (in Princeton's fall term, this is a gap of a whole month!) is enough time to start to put the entire course into perspective. On this time scale, the advice is not only reasonable, it is the only sound thing to do, if one operates according to the principle that no one ever learned a thing the first time he saw it. (Well, how could he?)

I want to convince you that Artin is super-cool. For that purpose there is at bottom only one thing to do: namely, show you that he's a rebel. A wild, wild rebel. Just thirteen pages into this book, not far into his appendix-at-the-beginning**, he has stated a theorem whose content is that when you fix a basis (of some vector space -- you're skipping that link, right?) there's a correspondence (“isomorphism,” in the vernacular) between linear transformations and matrices, (and change of the choice of basis corresponds to conjugation of matrices). He goes into a lamentation/screed for a page and a half, (emphasis added):
Mathematical education is still suffering from the enthusiasm which the discovery of this isomorphism has aroused. The result has been that geometry was eliminated and replaced by computations. Instead of the intuitive maps of a space preserving addition and multiplication by scalars (these maps have an immediate geometric meaning), matrices have been introduced. From the innumerable absurdities – from a pedagogical point of view – let me point out one example and contrast it with the direction description:

Matrix method: A product of a matrix A and a vector X (which is then an n-tuple of numbers) is defined; it is also a vector. Now the poor student has to swallow the following definition: A vector X is called an eigenvector if a number λ exists such that AX = λ X. Going through the formalism, the characteristic equation, one then ends up with theorems like: If a matrix A has n distinct eigenvalues, then a matrix S can be found such that S-1AS is a diagonal matrix. The student will of course learn all this since he will fail the course if he does not.

Instead one should argue like this: Given a linear transformation f of the space V into itself, does there exist a line which is kept fixed by f? In order to include the eigenvalue 0 one should then modify the question by asking whether a line is mapped into itself. This means of course for a vector X spanning the line that f(X) = λ X. Having thus motivated the problem, the matrix A describing f will enter only for a moment for the actual computation of λ. It should disappear again. Then one proves all the customary theorems without ever talking of matrices and asks the question: Suppose we can find a basis of V which consists of eigenvectors; what does this imply for the geometric description of f? Well, the space is stretched in the various directions of the basis by factors which are the eigenvalues. Only then does one ask what this means for the description of f by a matrix in terms of this basis. We have obviously the diagonal form.

I should of course soften my reproach since books have appeared lately which stress this point of view so that improvements are to be expected.

It is my experience that proofs involving matrices can be shortened by 50% if one throws the matrices out. Sometimes it cannot be done; sometimes a determinant must be computed.
He then re-enters the stream of the exposition. “Talking of determinants,” he says, “we assume that the reader is familiar with them.” And we're off.

But, by the by, and coming back to our concrete problem, is that prophecy correct, that future books will move toward Artin's geometrical view? I have a vast number of linear algebra books, and on inspection it's not so rosy. Some do exactly what Artin decries, without any apparent shame. Some even make linear transformations into second-class objects by introducing hideous notation for the matrix of a transformation with respect to such-and-such bases (not necessarily the same basis for the input as the output – good grief!). Some try to “motivate” the problem with differential equations, which from the point of view of an engineer may not be so ridiculous as it seems to us on the face of it (and anyway systems of first-order linear DE's are a classic application in such a course). Bretscher, the book they use here, is actually not so bad. It claims to emphasise geometry, and seems to do so pretty well, for the level of the class.

But there's the rub: Artin's approach is too difficult to teach to the students and still expect them to also master the matrix material they'll need to, you know, actually do some problems and not fail the course. In the end it is not a way to teach “something that's unlike anything [the students have] ever seen before,” because its emphasis on geometric character presupposes some geometric intuition to which one can appeal – in other words, some underlying familiarity not necessarily with linear algebra but, absent that, with some other and really more difficult mathematics. Bretscher's compromise, and it seems a reasonable one to me, is to give examples of matrices with special geometric meanings, transformations we've seen before (rotations, reflections, projections), and ask what their eigenvectors are. The students should be able to answer right from the geometry they already understand, without ever writing down a matrix, (although they could write it down if they wanted to, or felt the need to).

This balance between the conceptual and the formal the would-be instructor must maintain with care and deliberation. Maybe I won't stake my infant career on Artin's throwaway comments. I could just photocopy that page as a handout. If that handout wouldn't confuse anyone. But in that case I can always give it out to them, on my didactic authority if they don't feel comfortable judging it for themselves, that it's all right to skip it.


Endnotes.

* I am reminded of an English teacher from high school who wondered why it was, or how it came to be, that everyone thinks they should read a book by starting from page one and continuing to read page-by-page. Clearly there are many more ways to read the book, although most make no sense. Probably this habit is out of respect to the author, who presumably (though this belief is often well-characterised by the negative, skeptical connotation of "presumption") has put his industry and his learning into crafting a well-structured book. A reader inexperienced in some subject hasn't necessarily the knowledge to know what parts he needs to read to do whatever. But if all this is so it merely makes us wonder instead (1) why a well-structured book means a linearly-structured book; and (2) why more authors couldn't write such helpful "suggestions for the use of this book."

It is a custom, I should mention, in many corners of the textbook world to outline a couple of different options for the use of the book in a one- or two-semester class: cover these chapters but not those sections, and so forth.


** We need a good archaeologism, but for that one needs good Latin. “Precedix” is tempting, coming fairly directly from Latin “praecedere,” but doesn't carry quite the right meaning: it is “a thing coming before,” whereas Artin's appendix-at-the-front is more like elementary material. We could try “fundix,” from fundere, (cognate with to found: “found a city,” “a foundation,” and such), but it sounds ridiculous. Maybe “precedix” is better; after all, an appendix in English doesn't literally mean “a thing hanging on,” either.

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4 Comments:

At 12:57 PM, Anonymous Anonymous said...

Ignoring the math, if I may be so bold, there appears to be at the heart of your entire essay a thesis that the order of an idea doesn't matter as eventually the entire idea will be understood.

Elluding to you fiction example, I have read many a book that is either set in a variable time setting or written by a number of authors that could easily be read using random chapters and make as much sense as reading in the published order.

Referring to math as the example of ths day, how many times do you (the global metaphysical you) read a section or chapter only to realize a day or so later what was being said and the implications of that section. It could be old age but I think not.

Brains are a holistic memory storage and processing objects, and the full cross indexing and absorption takes time (a long time for some).

 
At 10:44 PM, Blogger BKF said...

Yeah. Except that as a pedagogue I couldn't say that "order doesn't matter," and not just for efficiency reasons. I complain that an ill-conceived structure us to the detriment of learning, insofar both as (1) it doesn't facilitate the putting of concepts into the order or structure they'll be in once learned; and (2) it is also a barrier even to the reader's making the attempt to learn the material. (This last isn't really an efficiency reason.) I can hardly stomach to read a book whose author hasn't put more thought into the writing than I would into the reading. I can only think that I should have more respect for my time and industry....

The presumption (almost always incorrect) that things should be presented in linear, page-by-page fashion -- totally ordered, in the jargon -- is one of these ill-conceived albeit prevalent notions. In practice I guess one tries to avoid it by repetition, recalling and updating old ideas at opportune moments, breaking discussions into parts, making independent and correctly delineated chapters, and so forth, so that the book takes on the character of several interwoven lines of thought. This reflects the more correct idea of a partially ordered collection of concepts. But an author can take the extra step to include a map or discussion of this structure underlying his book.

I think someone could make a very long-lasting newspaper column just pointing out situations in the world where there's some dubious presumption held that things are totally ordered.

 
At 7:36 PM, Anonymous Anonymous said...

As a marker and a soon to be teacher, there has to be a certain amount of entertainment value in your sessions. I believe it was Craig that commented on his sleeping during class as the professor was droning on about something that didn't create the spark that he needed to stay awake.

Quickly teaching a lesson and with a hearty ah-ha point out to the students how this ties back to the beginning of the class.

 
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