A Conversation with Antal Fekete – Real Linear Algebra

The following is the first in a series of interviews with Professor Antal E. Fekete.

The reader is encouraged to start here:

Professor Fekete, thank you for agreeing to this interview. As a long-time student of yours, I’m honored (which, I suppose, makes me an honored student.)

In this first installment I would like to discuss everyone’s favorite subject: mathematics.

Now gentle readers, before you stampede for the fire exits, recognize there are good reasons why I’ve chosen to being this conversation with mathematics.

  1. Fekete is a professional mathematician, and if we are to know anything about the man behind his incredible insights into gold dynamics and the Greatest Depression through which we are now struggling, we must understand from where he has come. And he has come from the mathematics department.
  2. Despite being a mathematician, Fekete, unlike the proverbial fellow with a hammer who only sees a world bristling with nails, understands the limits of mathematics, and refuses — unlike armies of economists — to be lured into applying mathematics to monetary economics where it is neither applicable nor ethical. There is much to be learned in this.
  3. His innovations in mathematics are profound in their own right, and well worth a peek for the novice, and some serious pondering for the pro.
  4. Mathematics undergraduate and graduate students should seriously consider mining Fekete’s work for superb deep thesis ideas. They are hanging low, ripe for the picking!


From Fekete’s curriculum vitae we read:

Antal E. Fekete, Professor, Memorial University of Newfoundland, was born in Budapest, Hungary, in 1932. He graduated from the Loránt Eötvös University of Budapest in mathematics in 1955. He left Hungary in the wake of the 1956 anti-Communist uprising that was brutally put down by the occupying Soviet troops. He immigrated to Canada in the following year and was appointed Assistant Professor at the Memorial University of Newfoundland in 1958. In 1993, after 35 years’ of service he retired with the rank of Full Professor.

Professor Fekete is the author of a book Real Linear Algebra and half a dozen papers on mathematics. In preparation are his monographs Quotient Set Theory and Stepnumbers.

In future installments I would like to delve into Quotient Set Theory and Stepnumbers, and their remarkable links with Fekete’s innovations around:

  • a new Braille-type code designed to help blind students become first-rate mathematicians;
  • digital transliteration of Chinese characters;
  • the world’s most efficient number system and quantum computing;
  • a new computational model for how the brain might work;
  • and, while we’re at it, putting all of mathematics on a new, more solid foundation to make it free of antinomies.

(And you thought mathematics was just a means to attract the opposite sex.) But for now we will begin with linear algebra. Let’s get real.

* * * * *

Professor Fekete, can you please share with us how and why you became interested in mathematics? I understand that Hungary has produced many superb mathematicians. Why is that?

Hungary was exceptional in the second half of the 19th and the first half of the 20th century in that, by a quirk of fate, there were lots of brilliant math teachers at all levels. Hundreds of schools in Hungary competed in graduating thousands of outstanding mathematicians. No longer.

The problem today, in Hungary as elsewhere, is that an educational bureaucracy has taken over from dedicated teachers. This bureaucracy has interest neither in searching for budding mathematical talent, nor in grooming talent if it sprouts spontaneously. It has absolutely no interest in teaching young people to think for themselves, for which mathematics is so superbly qualified. It has been ordered to train technicians how to turn the crank. That’s all that is needed in the numerous government research institutes in support of central planning and of military weapon system designs.

Most people I know have retained virtually nothing from their mathematics education. And this includes engineers and the like! You’ve taught for eons; what’s the problem? Is it the students? Is it the teachers? Is it mathematics itself? 

It is definitely the teachers. Their job has been reduced to that of a drill sergeant. They have neither inspiration nor love for the subject they are teaching. They hold their job to earn a paycheck while they enjoy extra long holidays and summer vacation. If there are a few exceptions, they are not held in high esteem for their devotion to their students and for loving the subject they teach.

One subject that virtually everyone in any field of science needs to have a command of is linear algebra. You chose to write the textbook Real Linear Algebra. Why linear algebra vs the infinite other possible subjects? 

The real number system ℝ is prototype for an overwhelming part of mathematics. The obvious question to ask is how to extend the one-dimensional model to higher dimensions, in particular, to dimension three, to get ℝ3. This extension is real linear algebra.

Why is linear algebra important (beyond the standard view), and more specifically, why should math teachers and students be interested in your text over the many others?

Teachers, colleges and universities all over the world are teaching the cross product of vectors ab in ℝ3. Recall that the cross product has all kinds of weird properties: it is nilpotent:

aa = 0;

it fails to be commutative:

ba = − ab.

It even fails to be associative; instead, it satisfies the Jacobi identity:

(ab) ⨯ c + (bc) ⨯ a + (ca) ⨯ b = 0.

It is a crazy kind of product. Where on earth has it come from? Where does it lead to?

Well, the cross product of vectors in the three-dimensional real vector space ℝ3 is just the Lie-product of the Lie algebra Lℝ3. [Lie is pronounced “lee”.] Lℝ3 can be described as the tangent space to the Lie group Oℝ3 of rotations of the three dimensional geometric space (which we visualize as a curved differential manifold, with tangent space of the same dimension having zero curvature). Moreover, there is a distinguished projection (also called by its technical name: functor) from the manifold to the tangent space:

Exp: Oℝ3 ⇾ Lℝ3

which to a rotation α ∈ Oℝ3 assigns the vector

Exp(α) ∈ Lℝ3: the direction of the rotation α. If α and β are rotations and αβ is their composition, then

Exp(αβ) = Exp(α) + Exp(β).

Moreover,

Exp(α−1) = − Exp(α).

Without this motivation linear algebra is an empty shell. Lie groups and Lie algebras are valid objects to study under the headings “algebras” and “groups”. The motivation for this study is the cross product of real linear algebra. The geometric images that come along are compelling and lasting. It is a scandal that this connection is kept from sophomore students and, even more so, from graduate students.

Another disaster area in teaching mathematics, next to the teaching of linear algebra, is advanced (multidimensional) calculus.

Yes. Here we find the worst textbooks ever written. There is only one exception: Advanced Calculus by Nickerson, Spencer and Steenrod, published in 1959 by Princeton University Press. It is a superb textbook, by now completely forgotten.

In that book the definite integral and multi-dimensional differentiation are put into proper context.  Then the definite integral of Calculus 100 appears as a special case of Stoke’s formula (!)

= f(b) − f(a) (definite integral)

c , dα > = < ∂c , α > (Stoke’s formula).

You would have never guessed it: the definite integral is just a bilinear scalar product < , > and differentiation d is a linear operator with its adjoint: d* = , the boundary operator so that ∂c is the boundary of the differential manifold c. (For example, if c is the northern hemisphere, then ∂c is the equator).

In general, if c is n-dimensional, then ∂c is (n – 1)-dimensional. Dually, if the differential form α is of rank n, then is of rank (n + 1).

The boundary operator ∂ satisfies ∂2 = 0 (the boundary has no boundary). Dually, d2 = 0 (the derivative of a differential form is always an exact differential form. We say that the differential form α is exact if = 0.)

Why is the definite integral a special case of Stoke’s formula? Because in that case c = [a, b], the interval over which integration takes place, with endpoints a, b which form the boundary of c in c. Note that c is 1-dimensional while ∂c is 0-dimensional. Dually, f(b) – f(a) is of rank 0 while df  is of rank 1 considered as a differential form.

Professor, this is a truly remarkable insight, looking at multiple integrals as a bilinear scalar product! Defining integration and differentiation simultaneously in a single formula is a magnificent unifying idea. (I sure wish this 

Indeed it is, as opposed to the fragmentation as they are treated in the run-of-the-mill accounts of the subject. The graded algebra of differential forms with the differential operator d joining the subsequent grades gives rise to De Rham cohomology.  The dual is the graded algebra of differential manifolds with the boundary operator ∂ joining the subsequent grades, giving rise to ordinary homology.

I hurl criticism to teachers of advanced calculus: they never ever point these things out to their students. Perhaps the reason is that they are ignorant of these facts themselves. It is not their fault. Their teachers were just as ignorant. They are teaching a bewildering variety of differential operators: grad, curl, div, ∆, without ever inquiring how they are related to the grand-daddy of all differential operators, d. The whole edifice of advanced calculus is falling apart for lack of any logical cohesion. Poor teaching, that’s what it is.

[The interested reader is encouraged to read Fekete’s blog post Putting Differential and Integral Calculus into Context.]

As the author, are there any other thoughts you might like share about Real Linear Algebra?

I can also characterize contemporary teaching of mathematics as a mad rush to generalize, from the real vector space ℝ3 to vector spaces over abstract fields, never pausing to take an unhurried look at the real vector space ℝ3 in which the major part of our lives is being played out. Sometimes I describe this process, tongue in cheek, as rushing to generalize the concept of a group to semigroups, to pseudo groups, to pseudo semigroups, to quasi-groups, to quasi-semigroups, to quasi-pseudo-semigroups, etc., ad nauseam. This is not a condemnation of studying semigroups. Rather, it is a criticism of “l’art pour l’art” generalization in order to flaunt one’s being state of art. It is not appropriate as a first exposure of students to fundamental mathematical ideas.

My answer to the race to stratospheric generalizations in the rarefied atmosphere of the abstraction of the abstract is: Real Linear Algebra.

And finally Professor Fekete, you hold the late mathematician Norman Steenrod in the highest regard as a mathematics teacher? Why? What made him special?

I had better let Steenrod himself speak on this.

[Below are excerpts from the preface to Real Linear Algebra, by Antal E. Fekete, (Chapman & Hall Pure and Applied Mathematics), hardcover, 1985, reprinted here with the author’s permission.]

In 1974 Professor D. C. Spencer of Princeton University invited me [Antal Fekete] to spend a year at Princeton and he put at my disposal that part of the mathematical archives of his friend, the late Norman Earl Steenrod, which had to do with sophomore mathematics courses. I found a veritable treasure trove in those archives. … The archives revealed that Steenrod had started collecting material for a naive linear algebra textbook, shifting the emphasis from algebraic rigor to geometric intuition. … While at Princeton I decided I would complete his unfinished task.

Why this is an important task is best told in Steenrod’s own words, spoken in 1967…

Although geometry pervades all mathematics and is present at every stage of development, too often we fail to point this out to our students. We rely on analytic formulations since we realize that they are complete and we are in a hurry to get on. We do not take time to look at geometric formulations. We are too greatly impressed by the rigor of analysis. We seem to feel that geometry is not rigorous, or at least that the background needed for rigor is not available. We feel that it is better not to do anything that is not rigorous. When we do present geometry, it is often the instructor who does the geometry while the student is merely a passive spectator. We present geometry to him in order to explain the analysis, but then we require him to do only the analysis — no geometry. We tend to avoid geometric formulations of questions in examinations. Questions are hard to formulate geometrically. Almost every time you try such a question, you find that a large group of students misinterpret it. Such questions are hard to grade because they are so varied. The absence of geometric questions on final exams tends to degrade the geometric content of the course, and leads to its neglect.

What has bothered me through the years is the control the exam seems to have on the course. Somehow the tail wags the god. In the exam we are supposed to take a sample of what the student knows. This process of sampling has a feedback effect that is very serious. The most famous example of this is the College Board exam and its influence on the teaching of mathematics in secondary schools. The examiners, in order to be fair to students in all parts of the country, tended to take the intersection of the topics taught in various schools. In the 1920’s and 1930’s the exam had little effect on the teaching of mathematics, but by the early fifties the feedback effect became pronounced. A greater number of students were taking the exams, and schools were rated by the results. If a particular high school had a poor rating, they did something about it: they compared carefully what they were teaching with the kinds of questions asked on the exams; they altered their curriculum accordingly, and concentrated on topics of maximum frequency. The examiners, on their part, observed the shrinkage and narrowed the range of questions accordingly. At one time it was projected that after forty years only one topic would survive this elimination process, and that would be the factoring of quadratics.

Some say this cannot happen in college because the instructor is in charge of his course. Well, he is not, because in many colleges there are freshman courses with large enrollments and many sections. To avoid troubles with young instructors giving wide varieties of grades we insist on uniform exams and uniform grading. I have seen the feedback effect time and time again while teaching a section of the freshman course. Along comes a bright fresh Ph.D. teaching his first class. Knowing that the concept of limit is central to calculus, he settles down and does a good job of teaching limits for two months. His students do very well on that one question, but not so well on the other four of a more routine nature. The average for his students is ten point below the overall average, so he finds himself giving D’s to students he thought were pretty good.Having learned his lesson, he runs a statistical analysis on the final exams for the last five years, and starts teaching his students how to turn the crank. By the end of the semester he normally brings up their average up to where it should be.

I do not know how to defeat this, but I do have one suggestions to offer. Harness the feedback effect to upgrade geometry by putting more geometric questions into their final exams and then face the problem of grading them. If, in the earlier parts of the course, on the ten-minute quizzes and the homework, you have inflicted geometry on the students over and over again, then on the final exam you have some chance of getting a good reaction out of the geometric questions.

This is, of course, not the only, nor the main reason why to bother with geometry at all The main reason is that most problems are presented in geometric form in the first place. Reformulation and solution in analytic terms is merely a second step. To complete the process, there is an indispensable third, namely the interpretation of the analytic solution in geometric terms. There is another reason, which is psychological. Two views of the same thing reinforce one another. Most of us are able to remember the multitudinous formulas of analysis mainly because we attach to each a geometric picture that keeps us from going astray. Even better than that, the geometric view of a problem helps us to focus on the invariants and to weed out the irrelevant details. A poor choice of coordinates may lead to a horrible mess in the analytic formulation, but with some geometric insights we may be able to choose the “best” coordinate system.

Now we come to the question of textbooks. The situation here is definitely sad. Books on linear algebra are written by algebraists for algebra students.  Of course, this is not in itself a condemnation, but there are two features that show up from this. One is that geometry is presented in an offhand fashion, if it is treated at all. Secondly, most textbooks confuse the presentation of theory with techniques of computation via matrices. One way to see that a book makes this mistake is to see that it has a chapter on determinants and matrices before linear transformations are defined. The linear transformation is an easy conceptual thing to talk about and give examples of without matrices. The matrix is a tool for computation, i.e., it is a set of coordinates subject to a prior choice of basis. The matrix, important as it may be for computation, is of no importance in the theoretical or conceptual part of the course, nor in the geometric pictures which come along. Presenting semi-theoretical material on matrices, and teaching the students to work with matrices before introducing linear transformations, is comparable to trying to teach someone to play the piano on a keyboard that isn’t attached to any strings. There is no feedback, the student does not see the objective and finds no pleasure in what he is doing. True, historically matrices came first. For a long time a vector space was an ℝfor some n and a linear transformation was a system of linear equations represented by the matrix of coefficients. Thus the properties of linear transformations had to be formulated as properties of matrices. In this way an extensive theory of matrices arose. It is a cumbersome theory both in notation and conception. The conceptual point of view, that one could proceed on a different level and work without coordinates, developed during the twenties and thirties. It became clear that matrix theory tended to obscure the geometric insight. With the new point of view the picture became quite easy and lovely, and the theory was disassociated from the mechanism of computation. Thus it is easy to see why the first books on linear algebra had to begin with matrices and determinants, but it seems to me that the conversion to the more recent and simpler view has been much too slow. It should be clear to the students that matrices are not essential to understanding the theory and that the theory must not be confused with the computations which arise.

Another inadequacy of many texts is that the structure theorems for linear operators are usually only given in the complex case. This case is algebraically easier and smoother because the characteristic equation splits into the product of linear factors. The details of the real case are omitted, in spite of the fact this is the case of interest because of the geometry, and it should be studied carefully before one proceeds to the complex case.

As the title Real Linear Algebra suggests, it takes an unhurried look at the vector spaces of the real numbers, in particular, at the three-dimensional real vector space ℝ3 in which a major part of our lives is played out. It highlights the remarkable fact that the cross product converts ℝ3 into a noncommutative Lie algebra that is compatible with the metric — an exclusive feature restricted to n = 3. Moreover, the Lie algebra is isomorphic to the Lie algebra of antisymmetric operators and, therefore, it is the Lie algebra of the group of rotations in three-dimensional space…

The isomorphism between the algebra of linear operators and the algebra of matrices is put on the same footing as the isomorphism between the n-dimensional real vector space V and ℝn, the vector space of coordinates…

The geometric background to symmetric, antisymmetric, and orthogonal operators is provided. Of these, the antisymmetric case is omitted from most accounts. This oversight can hardly be condoned in view of the extraordinary importance of the Lie algebra of antisymmetric operators which give rise to the orthogonal group under the exponential functor, and so it is the only reasonable candidate to generalize the cross product to higher dimensions…


Norman E. Steenrod (1910 - 1971)