# Manual On the Teaching of Linear Algebra

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.

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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.

## When is a good time to teach linear algebra? - Mathematics Educators Stack Exchange

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. A Little History 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. May in the s, even before Sheldon Axler  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 s. Particularly influential in this process were the book of B.

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Both were on "Modern Algebra," but included chapters on linear algebra. Historian Jean-Luc Dorier  regards Paul Halmos' book  Finite Dimensional Vector Spaces, first published in , 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 and 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 MacLane taught the same course see  , page 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 s and 60s and some colleges and universities were still adding the course in the early s. It appears that the linear algebra course I had in 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.

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  regards the axiomatization of abstract vector spaces to have been completed in the s 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 s, few engineers could hope to solve a system of equations in unknowns; linear algebra was really irrelevant! By the s engineers were beginning to use computers to solve practical problems using linear algebra. For example, in , 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 X matrix that were close to Unfortunately, at the time, I had no clue-the best advice I had to offer was to find all 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.

I think you should not study linear algebra if you are just getting started with applied machine learning. I call this approach a results-first approach. It is where you start by learning and practicing the steps for working through a predictive modeling problem end-to-end e. This process then provides the skeleton and context for progressively deepening your knowledge, such as how algorithms work and eventually the math that underlies them.

## Reflections on fall 2018

Linear algebra is the mathematics of data and the notation allows you to describe operations on data precisely with specific operators. Further, programming languages such as Python offer efficient ways of implementing linear algebra notation directly. An understanding of the notation and how it is realized in your language or library will allow for shorter and perhaps more efficient implementations of machine learning algorithms.

A challenge for newcomers to the field of linear algebra are operations such as matrix multiplication and tensor multiplication that are not implemented as the direct multiplication of the elements of these structures, and at first glance appear nonintuitive. Again, most if not all of these operations are implemented efficiently and provided via API calls in modern linear algebra libraries. An understanding of how vector and matrix operations are implemented is required as a part of being able to effectively read and write matrix notation.

You must learn linear algebra in order to be able to learn statistics. Especially multivariate statistics.

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Statistics and data analysis are another pillar field of mathematics to support machine learning. They are primarily concerned with describing and understanding data. As the mathematics of data, linear algebra has left its fingerprint on many related fields of mathematics, including statistics. In order to be able to read and interpret statistics, you must learn the notation and operations of linear algebra.

Modern statistics uses both the notation and tools of linear algebra to describe the tools and techniques of statistical methods. From vectors for the means and variances of data, to covariance matrices that describe the relationships between multiple Gaussian variables. The results of some collaborations between the two fields are also staple machine learning methods, such as the Principal Component Analysis, or PCA for short, used for data reduction. Building on notation and arithmetic is the idea of matrix factorization, also called matrix decomposition.

Matrix factorization is a key tool in linear algebra and used widely as an element of many more complex operations in both linear algebra such as the matrix inverse and machine learning least squares. In order to read and interpret higher-order matrix operations, you must understand matrix factorization. Linear algebra was originally developed to solve systems of linear equations. These are cases where there are more equations than there are unknown variables e. As a result, they are challenging to solve arithmetically because there is no single solution as there is no line or plane can fit the data without some error.

Regular homework assignments must be completed. These assignments help students solidify concepts in their minds by practicing using important theorems and ideas. Students are also asked to extrapolate and apply ideas to new situation in homework. This practice is essential to understanding the material, and students who do not keep completely up-to-date with homework often find themselves very far behind in understanding. Students are also required to take careful, complete class notes since classroom presentation may deviate from the order of topics in the text, present alternate approaches to the material, or even present material not contained in the textbook.

Students are responsible for all material both from the text and presented in class. Students will want to form study groups or meet outside of class either with or without the teacher in order to discuss and familiarize themselves with the material. Students are evaluated on the quality of their written homework and on exam performance. Course Resources: homework, handouts, etc.