# On the Centrality of Linear Algebra in the Curriculum

## Carl C. Cowen

*Remarks on receiving the Deborah and Franklin Tepper Haimo Award for
Distinguished College or University Teaching of Mathematics, San Diego
California, January 1997.*

While this is an opportunity for me to lay out my "secrets" for successful
teaching, I have too few secrets and they are too well known for me to talk
for more than a few minutes about them.
The first step for a person to take toward good teaching is to recognize
that the pursuit of teaching excellence is a worthy goal. Since I believe
that good teaching is important, I'm very glad that today this first step
is getting easier for many people to make. Having decided that teaching
well is important, the next step is to think about the issues involved and
to educate oneself about the things that work for other people. I'm very
grateful to my colleagues who have modeled good teaching, answered my
questions about approaches to a lesson or a subject, and encouraged me in
my efforts to teach more effectively. Some of the people who deserve my
thanks are Bill Fishback and Harold Hanes at Earlham College and Guershon
Harel, Jim McClure, J. J. Price, and Bob Zink at Purdue. I have learned a
great deal about teaching mathematics from talking with them, but I have
learned even more about teaching from talking with my wife Janice, who is
an exceptional Spanish teacher. Finally, it is important to know the
students, to talk to them and, especially, to listen to them. Get to know
their names, get them to talk in class and ask questions, and get them to
talk outside of class, too. Students will tell you, although not always
directly, what they don't understand and then you can help them develop
their own answers to their questions.

If you know the students' names, you can say hello to them in the hall and
you can ask them, by name, to answer a question in class. While this is
intimidating, if you ask every student in the class a question every week
or two, they will quickly learn that they aren't being picked on. Generally
I don't take "I don't know" as an answer to a question, it just leads me to
ask a simpler related question that will help them discover the answer to
the question I originally asked. When students are in the habit of speaking
in class, they will more readily ask questions and if you routinely ask
them questions, they are more likely to ask questions of you before you ask
them the question they can't answer. Of course, how you handle questions is
critical. You might be fortunate enough that all their questions are
insightful and lead you on to the next topic, but don't count on it. You
must take pains to regard every question as serious and deserving of a
thoughtful reply. Students will only ask questions if they are reasonably
comfortable doing so and your indication that any question is fair will
help them be comfortable about asking questions even when they can't tell
which questions are "dumb." But just because someone asks a question
doesn't mean I give them a straight and simple reply! I'm much more likely
to regard a question as a starting point for a discussion of what they
understand and don't understand about the situation.

Instead of talking more about my secrets, I want to talk at some length
about the role of linear algebra in the curriculum and the opportunities it
presents for teaching. A good friend of mine tells me that teaching
differential equations is much more rewarding and interesting; she is
probably right, for her. For me, though, linear algebra has become the
focus of my instructional work and having that focus has been critical to
my development as a teacher; I urge you to find the venues that are right
for you to develop your teaching skills and that will enable you to make
the strongest contributions in your institution.

I taught linear algebra the first semester I was in a college classroom and
most semesters since then. At first this was accidental, but later, as I
began to consider the curricular issues involved, I realized that linear
algebra plays a pivotal role in the curriculum both for majors and
Mathematically oriented non-majors, so I wanted to teach it frequently.

There is a tendency to believe that the structure of mathematics and, with
the exception of calculus reform, the college mathematics curriculum have
remained the same for a very long time. This is far from true. In fact,
linear algebra, as we know it today, has existed a comparatively short
time.

My first realization of the changes that have taken place came from seeing
the MAA film Who Killed Determinants? (done by Kenneth O. May in the 1960s,
even before Sheldon Axler [1] decided it was a good idea
they be killed). May documented how determinants flourished in the 19th
century with its connections to the study of invariants and how the study
of determinants developed into the linear algebra we know today, where
determinants are far from central. Linear algebra did not really come to be
recognized as a subject until the 1930s. Particularly influential in this
process were the book of B. L. van der Waerden [9] from
1930 to 1931 and the book of Garrett Birkhoff and Saunders MacLane [2] of 1941. Both were on "Modern Algebra," but included
chapters on linear algebra. Historian Jean-Luc Dorier [5]
regards Paul Halmos' book [6] Finite Dimensional Vector
Spaces, first published in 1942, as the first book about linear algebra
written for undergraduates. This is all much more recent than I would have
guessed a few months ago!

In 1936 and 1937 at Harvard, Birkhoff taught an algebra course that
included an axiomatic treatment of vector spaces over a field and linear
transformations on finite dimensional vector spaces, and during 1939-40
MacLane taught the same course (see [8], page 295). In
preparation for this talk, I looked at catalogs from several colleges and
universities to find out when their first undergraduate courses in linear
algebra were taught. The separate linear algebra course became a standard
part of the college mathematics curriculum in the United States in the
1950s and 60s and some colleges and universities were still adding the
course in the early 1970s. It appears that the linear algebra course I had
in 1965 at Indiana University was one of the first times it was offered
there as a regular course although, at the time, I thought math majors had
been taking it for decades. The catalogs make it clear that linear algebra
courses had been split away from the abstract algebra courses that had
developed earlier. This was reflected in the very abstract nature of the
courses many of us took then: indeed, I could prove theorems on
determinants of linear transformations on an abstract vector space but
would have had difficulty in finding the determinant or inverse of a 4X4
matrix!

Thus, in the past forty years or so, the linear algebra course has come
into being as an abstract course for serious majors, has been revised into
a first "intro to proof" and "intro to abstract mathematics" course for all
math majors, and in many places has now become a sophomore matrix-oriented
course for a wide variety of majors.

Why am I telling you this? I want you to realize that (regardless of the
prevailing attitudes in your department) linear algebra has not "always"
been done the way it is now, to suggest that we are in the middle of a
"reform," and to use the history of the reform so far to point out where I
think we are and should be going.

The first step is try to understand the developments up to now. I believe
that the first courses grew out of the general axiomatic approach to
mathematics that was common at that time. Historian Gregory Moore [8] regards the axiomatization of abstract vector spaces to
have been completed in the 1920s and many areas of mathematics had their
foundations developed in the first third of the century. I think the
success of the axiomatic method in this and related algebraic areas, as
well as the basic and important mathematical content, contributed to
abstract algebra and linear algebra being given a prominent place in the
curriculum first for serious majors then for all math majors.

But the more recent phase of the reform has a different origin: I believe
it is due to the development and widespread use of the computer in areas
that apply mathematics. Surely engineers have known for more than a century
that many problems could be modeled as systems of linear equations or as
eigenvalue problems. But what would be the point? Even in the 1950s, few
engineers could hope to solve a system of 100 equations in 100 unknowns;
linear algebra was really irrelevant! By the 1970s engineers were beginning
to use computers to solve practical problems using linear algebra. For
example, in 1974, a graduate student friend studying civil engineering and
working on modeling vibrations in buildings caused by earthquakes asked me
how he could find the eigenvalues of a 200X200 matrix that were close to
12. (Unfortunately, at the time, I had no clue-the best advice I had to
offer was to find all 200 and check which were closest to 12; I know better
now!) In the past two decades, the applications of linear algebra to real
world problems have mushroomed. The computer software *Matlab*
provides a good example: it is among the most popular in engineering
applications and at its core it treats every problem as a linear algebra
problem. Suddenly students from all over the university are being advised
to take a linear algebra course. The influx of these students with their
different interests and, with the higher percentage of the population going
on to college, the influx of students who are mathematically not as well
prepared have forced many colleges and universities to change from courses
dominated by proofs of theorems about abstract vector spaces to courses
emphasizing matrix computations and the theory to support them.

The change in the audience for our linear algebra courses creates the
necessity, and the opportunity, to reexamine our approach to teaching the
subject. After all, the essence of teaching is to help students learn the
material that they need and want to learn: with different students wanting
to learn different material, we should expect to change our teaching.

In thinking about teaching this subject, the first realization that we must
come to is that linear algebra is incredibly useful in the modern world,
probably more useful than any other college level mathematics with the
possible exception of calculus. Several of my former students have told me
that linear algebra was the most useful math course they took in college
and can give specific examples of why they say that. Students who are in
the courses now believe this is the case and most new linear algebra books
include applications that look to students like they are non-trivial. While
I don't believe this should force us to teach the applications, I do
believe it necessitates our being conscious of the applicability and the
change that requires in our teaching style. I don't believe it forces us to
abandon theory, but it should encourage us to see the role of the theory in
the subject as it is applied.

The second realization we must come to is that no serious application of
linear algebra happens without a computer. This must change the nature of
the course; I believe it strongly argues for including computation as a
part of the course. Fortunately many calculators can do all the
computations that arise in a first course and software such as
*Matlab*, *Maple*, and *Mathematica* can do all that and
more. We need to be at least vaguely conscious of the ways computers do
linear algebra and how that affects the way we approach the subject. For
example, the standard approach to eigenvectors and eigenvalues, and for
years the only one I knew, was to find the characteristic polynomial of the
matrix, find the roots of the polynomial, and solve the eigenvector
equations for each eigenvalue. This approach is hopeless for
practical-sized matrices! Indeed, for large matrices, it is difficult to do
even the first step of finding the characteristic polynomial. Nevertheless,
the relationship between the characteristic polynomial and the eigenvalues
is still important for students to understand and the QR-algorithm, a
numerical approach for finding eigenvectors and eigenvalues, is not
appropriate material for a first course in linear algebra. But, in spite of
this, our students should not leave our courses thinking the characteristic
polynomial is the only way to find the eigenvalues of a matrix.

What I am really arguing for is an effective integration of computing into
the classroom. Although I don't know what that is in every circumstance,
leaving the computer out of the classroom is certainly not the most
effective approach. Many faculty around the country are working on how to
incorporate computing effectively and the answers that are being developed
are sure to change the way our classrooms will look, how we will teach, and
ultimately, how our students will learn. One point that deserves mention is
that math majors tend to be less computer literate than some other majors,
a disadvantage in the 1990s. If we integrate computing into the math
majors' courses, they will graduate with more confidence and more useful
computer experience, and I believe we will therefore be able to attract
more majors.

At the surface level, many faculty agree with the point that after the
basic hand calculation techniques have been mastered, having students use a
machine to do the arithmetic is very helpful. For one thing, the students
can concentrate on this week's ideas instead of trying to get the
arithmetic right in the solution of the relevant linear system.

At a deeper level, though, if the instructor is armed with a computer and
display device so that the class can see her or his work, the instructor
can do things that are not otherwise possible in class. For instance, in
working an example, I like to ask the class how to approach the problem. If
I am working strictly with a blackboard and prepared-in-advance
calculations, when a student proposes an inappropriate approach to the
problem, I will not have those calculations pre-prepared and will be
unwilling to take class time to do them at the blackboard: I am left
explaining, perhaps unconvincingly, why the suggested approach is
inappropriate. On the other hand, armed with a computer, I am usually
willing to do exactly what the students tell me to do: when the misguided
attempt fizzles, they see it first hand rather than simply taking my word
for it, and they are motivated to participate in a second attempt. One of a
teacher's roles is to demonstrate thought processes. Too many times, a
lecturer seems to be omniscient, always knowing exactly how to do every
problem. Students aren't like that-they occasionally make mistakes and they
need to learn how to recognize them as mistakes and recover. You and I know
that we are not perfect in our offices, but the students don't. We see when
we have made a mistake and we have learned how to start again with a new
approach-our students will benefit from seeing us recover from a failed
attempt.

In addition, I am more willing to check the result of a calculation if I
have a computer because it is fast. For example, if the question asks us to
split a vector z into two pieces z = w + u in such a way that w is in the
subspace M and u is orthogonal to it, checking provides a mental review of
the problem as we are punching the keys to check that z = w + u , w is in
M, and u is orthogonal to M, and they can rethink what the conditions
mean. My colleague Jim McClure likes to start a calculation and then ask
what will happen when he pushes the "return" key. He finds students are
more willing to participate when he is using the computer than if he does
the computing at the board. I suspect students can imagine themselves in
the situation looking at the computer screen more easily than across the
desk from the instructor.

Some faculty worry that using computers in a math course will turn the
students into unthinking button-pushers. While this could be true in some
circumstances, it is easy to avoid in linear algebra. Indeed I think the
computer can be used to motivate learning the theory and to reinforce the
concepts. I believe that in linear algebra, more than in any other
elementary college mathematics course, the theory plays an essential role
in computations and that use of the computer can make this evident. A good
example is solution of linear systems. The theorem engineers call the
"Principle of Superposition" says that every solution of a linear system
can be written as the sum of a particular solution of the system and some
solution of the related homogeneous system. In the past, I have found this
theorem to be difficult for students to understand and
appreciate. *Matlab* has a command "" that gives one solution to any
system, namely the least squares solution, and another command "null" that
gives an orthonormal basis for the nullspace of a matrix. The software
provides both a reason to understand the theory and a mechanism for using
the theorem to easily find the general solution of a system of
equations.

In addition to examples of solving problems, computers make possible "gee
whiz" classroom demonstrations that can stimulate the students' geometric
intuition. I regularly use *Matlab* "movies" to demonstrate linear
transformations of the plane and their eigenvectors. I also have used
similar movies to demonstrate the geometric meaning of the Singular Value
Decomposition in my graduate class for engineering students. Right now,
Roger Lautzenheiser, of the Rose-Hulman Institute of Technology, and his
student Brad North are putting the finishing touches on a demonstration
package for linear algebra courses. The package is called Visual Linear
algebra. It runs under *Matlab* and has a nice enough interface that
students can play with the geometry of linear transformations and their
ranges and nullspaces and can see the rank-nullity theorem in
action. Certainly their demonstration at last fall's Indiana Section
meeting elicited lots of "gee whiz's!" The package is free and can be
downloaded from
http://www.rose-hulman.edu/~lautzenh/vla.html.

In addition, the computer makes it possible to ask questions involving
theoretical issues that are arithmetically too complicated to ask a student
to do with pencil and paper. For example, many students believe that every
subspace has a special, preferred basis and do not really understand the
implications of the fact that every subspace has infinitely many bases. A
question I like to ask students to do on the computer is the following:

John and Mary are taking linear algebra. One of the problems in their
homework assignment was to find the nullspace of the 4X5 matrix A. John's
answer was that the nullspace is spanned by (-2, -2, 0, 2, -6), (1, 5, 4,
-3, 11), (3, 5, 2, -4, 13), and (0, -2, -2, 1, -4). Mary's answer was that
the nullspace is spanned by (1, 1, 0, -1, 3), (-2, 0, 2, 1, -2), and (-1,
3, 4, 1, 5). Are their answers consistent with each other?

I think the question brings out the idea that there can be more than one
spanning set for a subspace in a context that is meaningful to the students
and I think it requires students to confront issues that are difficult for
them like, Which vectors are in a subspace? When are two subspaces the
same?

One approach to this problem is to try to write each of John's vectors as a
linear combination of Mary's and vice versa. The definitions and first
theorems about spanning sets say that if each of John's vectors is a linear
combination of Mary's vectors, and each of Mary's vectors is a linear
combination of John's vectors, then the subspaces they have described are
the same, otherwise they are not. By hand, this would be incredibly
tedious, but with a machine, it is a tolerable approach. A more
sophisticated solution is to find the rank of the 5X4 matrix whose columns
are John's vectors and then find the rank of the 5X7 matrix whose columns
are John's and Mary's vectors both. *Matlab* can find the rank of a
matrix in a single command, so this is very easy on the computer. On the
other hand, this approach requires students to really understand how the
column vectors are connected with the range of a matrix and how the
dimensions of nested subspaces are related to equality of the subspaces.

Many students do not have a good idea of how to start this problem and many
will try some approach and need to start over when they realize that it
will not work. The fact that the students have access to a machine makes me
willing to ask questions like this that may require several false starts
and a lot of arithmetic. I would typically assign this problem as homework
and after the class has struggled with the problem, talk about both
approaches, using the computer to demonstrate.

Reform in linear algebra is healthy and moving forward. In 1993 the Linear
Algebra Curriculum Study Group published a set of recommendations for
appropriate course syllabi for various kinds of classes. [3] While you may not agree with all of their
recommendations, you will surely find their work stimulating and thought
provoking as you design your own course. The ATLAST project has developed
and published a set of computer projects suitable for linear algebra
classes [7] and the MAA has published a book of resources
[4] useful in teaching linear algebra. Many colleges and
universities are introducing a second course in linear algebra because they
recognize the importance of the subject and the inadequacy of their first
course in meeting the diverse needs of the students. In addition, faculty
all over the country are rethinking the place of linear algebra in their
curriculum and the best approaches for teaching it. Indeed there were three
sessions at the San Diego meeting on innovations in teaching linear
algebra.

To summarize, I believe that linear algebra deserves a central place in the
curriculum of math majors, and other students as well, because it is widely
applicable, because it is a subject where students can see, even without
axiomatics, the development of a substantial mathematical theory, because
it is a subject that provides the opportunity for students to see the role
of that theory in doing computations and applying mathematics, and because
it provides a vital arena where students can see the interaction of
mathematics and machine computation. I believe that the integration of
computation and theoretical mathematics is so natural in linear algebra
that students (and faculty!) can use their experience with linear algebra
as a starting point for seeking similar integration in other mathematical
areas. Linear algebra provides a course that is full of ideas, with
material that is rewarding to learn and to teach, and is a subject where
both student and teacher can be challenged to their best
performance. Finally, I don't think students arriving as freshmen know how
to learn, certainly they don't know how to learn mathematics. Students need
to learn how to integrate a theoretical and computational understanding of
mathematics. Learning linear algebra can help them do that:

*Students who have learned how to learn linear algebra have learned how to
learn mathematics!*

I believe teaching is important and linear algebra is one of my favorite
places to use and improve my teaching skills. I am indeed grateful to be
recognized by the MAA for my efforts in teaching.

References

[1] Axler, S. "Down with Determinants!" American
Mathematical Monthly vol. 102 (1995): 139-154.

[2] Birkhoff, G. and MacLane, S. 1941. A Survey of Modern
Algebra. New York, NY: Macmillan.

[3] Carlson, D., Johnson, C. R., Lay, D. C., and Porter,
A. D. "The Linear Algebra Curriculum Study Group Recommendations for the
First Course in Linear Algebra." College Mathematics Journal vol. 24
(1993): 41-46.

[4] Carlson, D., Johnson, C. R., Lay, D. C., Porter, A. D.,
Watkins, A., and Watkins, W., eds. 1997. Resources for Teaching Linear
Algebra. Washington, DC: MAA.

[5] Dorier, J. "A General Outline of the Genesis of Vector
Space Theory." Historia Mathematica vol. 22 (1995): 227-261.

[6] Halmos, P. 1942. Finite Dimensional Vector
Spaces. Princeton, NJ: Princeton University Press.

[7] Leon, S., Herman, E., and Faulkenberry, R. 1996. ATLAST
Computer Exercises for Linear Algebra. Upper Saddle River, NY: Prentice
Hall.

[8] Moore, G. "The Axiomatization of Linear Algebra:
1875-1940." Historia Mathematica vol. 22 (1995): 262-303.

[9] van der Waerden, B. L. 1930-31. Moderne Algebra, 2
vols. Berlin, Germany: Springer Verlag.

Professor Cowen teaches at Purdue University in West Lafayette,
Indiana. His e-mail address is

cowen@math.purdue.edu.

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