More Linear Algebra, Please

By Drew Armstrong, Associate Professor of Mathematics, University of Miami

Anyone who teaches mathematics in the US knows that the quality of education could be better, but we also know that the problems are complicated and defy easy solutions. I grew up in Ontario, Canada, where I attended high school and completed an undergraduate degree in mathematics. Afterwards I completed a Ph.D. in the United States and I have now been teaching undergraduate mathematics here for over ten years. These experiences suggest to me a change that would improve college mathematics education in the US. It won’t solve every problem, but it is something concrete that we can do right now.

Suggestion: Replace the typical one-semester “introduction to linear algebra” course with a two-semester linear algebra sequence. This would be taken in the first year of college, in parallel with calculus. It would not have calculus as a pre-requisite.

In effect, this would place linear algebra and calculus side-by-side as the twin pillars of undergraduate mathematics. I believe this would have several immediate benefits for the curriculum. In this blog post I’ll describe three of these benefits and then I’ll explain how my experience as a student in Canada and as a professor in the US has brought me to this position.

Three Fundamental Problems

Linear algebra is being undersold. Linear algebra is the common denominator of mathematics. From the most pure to the most applied, if you use mathematics then you will use linear algebra. This is also a fairly recent phenomenon, historically speaking. In the 19th century, linear algebra was at the cutting edge of mathematical research. Today it is a universal tool that every user of mathematics needs to know. This becomes more true every year as algorithms and data play a bigger role in our lives. It seems to me that the current curriculum was fossilized in an earlier time when linear algebra wasn’t so useful. But times change; sometimes we need to re-examine the mathematics curriculum to see if it is still relevant. I believe that a two-semester linear algebra sequence in the first year will be a more honest representation of how mathematics is used today.

Complex Numbers. Complex numbers are currently an orphan in the undergraduate curriculum. According to the Common Core Standards, students are supposed to learn about complex numbers in high school; however, from my experience with US college students I know that they are not learning this material. At the University of Miami (where I teach), the basic ideas of complex numbers including de Moivre’s Theorem appear in our pre-pre-calculus remedial course, so it is not reasonable that a student who takes this class will have time to complete the math major in four years. Therefore, in practice, we are pretending that our undergraduates have seen complex numbers when this is not the case. It seems that many of our students are introduced to complex numbers for the first time in an upper-level complex analysis course. In my opinion this is way too late. If US students are not to learn about complex numbers in high school, then it seems to me that the first semester of a two-semester linear algebra sequence is an excellent place to introduce this material. Here they can see the 2-dimensional geometric interpretation of complex numbers via 2X2 rotation and dilation matrices. This would have the added benefit of teaching complex numbers to future high school teachers (who are usually not required to take a complex analysis course).

Linear Algebra remediation eats up other courses. At the University of Miami we require all math majors to take MTH 210 (Introduction to Linear Algebra). As is typical in many departments, this is a one-semester course that is usually taken in the second year, as it has Calculus II as a pre-requisite. Most of the students have never seen vectors or matrices before, so our goal is to get them from the basic ideas to the useful applications in one semester. This course is then required as a pre-requisite for many upper-division courses. However, most instructors find that the students’ subsequent linear algebra background is very shaky because one semester is not enough to absorb all of the material. In practice this means that many upper-division courses must include a crash course in the relevant ideas from linear algebra, which is hugely inefficient. A two-semester first year sequence would ensure that teachers of upper-division math courses could assume that the students are familiar with the basic ideas of linear algebra. This would save time and allow each of these classes to cover more material.

This Has Been Done Before

I attended high school and college in Ontario, Canada between 1993 and 2002. In those days we had 5 years of high school; the fifth year (called OAC for “Ontario Academic Credit”) was optional for the general population, but it was required for entrance to many university programs. The OAC mathematics curriculum consisted of three courses: Algebra and Geometry, Calculus, and Finite Mathematics.  A student entering a Bachelor of Science program at an Ontario university was expected to take all three of these. The Calculus course was similar to the AP Calculus course that exists in American high schools today. The Finite Mathematics course was an introduction to probability and statistics with a discussion of induction and elementary combinatorics. The OAC Algebra and Geometry course was a novelty that has no parallel in American schools. (Since the OAC program was phased out in 2003 it no longer exists in Canada either.) To give you an idea of the course, here is the table of contents from the final version of the course textbook (now out of print).

The curriculum was mostly analytic geometry in 2 and 3 dimensions providing a thorough introduction to representing lines and planes in 3D. It also introduced Gaussian elimination, orthogonality, the language of vectors with applications to physics, matrices as linear transformations, and complex numbers.  I remember this course very vividly as one of the experiences that led me to become a mathematician. Ontario universities required these OAC courses for entrance to certain programs. The undergraduate curriculum was then able to make good use of these courses. For example, when I entered Queen’s University as a mathematics and physics major I was required to take two full-year mathematics sequences in my first year. One sequence was Calculus and it required OAC Calculus as a pre-requisite. The other sequence was Linear Algebra and it required OAC Algebra and Geometry as a pre-requisite. Let me emphasize this situation:

I was required to take a semester of linear algebra in high school, which was then followed by a required two-semester linear algebra sequence in the first year of college.

The textbook that we used for Linear Algebra was the 3rd edition of “Elementary Linear Algebra” by Larson and Edwards. This text is indistinguishable from the standard textbooks used today for the “introduction to linear algebra” course in the United States. Thus, by this point I had seen the same material, but I had seen it over three semesters instead of one.

After the first year sequence was an additional one-semester course in linear algebra that was required for mathematics majors with a concentration in statistics or physics. (It was also required for many engineering programs.) This course gave a more abstract introduction to inner-product spaces, with applications to function spaces. The students in the course were diverse, coming from many different departments. This indicates to me that the material of this course was still seen as universal in the sense that it was relevant to all users of linear algebra, whether pure or applied. It was not a special topics course.

In summary, by the time I had completed a Bachelor of Science degree in mathematics with a physics minor, I had been required to take four semesters of general-purpose linear algebra. And you want to know something funny? As I proceeded to graduate school in pure mathematics at Cornell, I shortly came to feel that the most serious deficiency of my undergraduate education was that I had not seen enough linear algebra! Not only did I find my multilinear algebra background weak when I learned representation theory, I was also shocked when I learned about the Perron-Frobenius theorem and its amazing applications (e.g., to ranking webpages): Why had no one told me about this theorem?

My Teaching Experience

I joined the mathematics faculty at the University of Miami in 2009, and since then I have taught our Introduction to Linear Algebra course (MTH 210) four times. After this, many upper-division courses (such as Multivariable Calculus) list MTH 210 as a pre-requisite. We have two upper-division linear algebra courses in the curriculum (one more abstract and one more numerical), but these are regarded more as capstone courses and they do not serve as pre-requisites for anything else. Thus MTH 210 is a very important course.

Let me describe my experience teaching the course and how my approach to the syllabus has evolved. The official department syllabus reads as follows:

Vectors, matrix algebra, systems of linear equations, and related geometry in Euclidean spaces. Fundamentals of vector spaces, linear transformations, determinants, eigenvalues, and eigenspaces.

The majority of students in the course have never seen the language of vectors and matrices before, nor have they taken a physics course which could be used for motivation. In my opinion, it is completely impossible to cover the material from a standard textbook under these conditions, so my experience with the course has been a painful process of deciding what to leave out. After trial and error I have decided that the abstract notions of “vector space” and “linear transformation” had to go, so that I can get to some substantial applications by the end of the semester. The language of matrix arithmetic is already plenty abstract for students seeing it for the first time.

Since the students are starting from scratch, I begin the course with a thorough introduction to Cartesian coordinates and analytic geometry in 2 and 3 dimensions. Without the visual intuition that this provides I don’t think they’ll get very far. By the halfway point of the semester I want to cover the standard material on Gaussian elimination and to be able to explain the geometric intuition behind it. 

After covering these minimum pre-requisites in the first half of the semester, in the second half we dive into the actual “linear algebra”. Here is where the painful cuts come. Since this is the only linear algebra course that most of these students will ever see, I need to cover the significant applications. But I also need to cover enough theory that these applications can be understood. With practice I’ve come up with a way to do this that I’m reasonably happy with. If you want to see the details, my lecture notes from two previous iterations of the course are available here and here.  But on the other hand, I’m not happy at all. I know how useful linear algebra is, and I know that most of these students would benefit from a deep understanding of it. I also know that this one-semester class goes by too quickly for the ideas to really sink in. I can only hope that I’ve given them a roadmap so they can fill in the details later when needed.

Conclusion

It seems clear to me that a two-semester linear algebra sequence would vastly improve the undergraduate mathematics curriculum in the United States. This would give extra time to absorb the important ideas and give space to discuss neglected topics such as complex numbers and analytic geometry. It would also allow upper-division courses to use linear algebra in a deeper way.

This is natural from a mathematical point of view, but I also recognize that from an administrative point of view it’s pretty radical. I’ve sat on committees and I know how time consuming it can be just to change the title of an existing course. Adding a new course at the center of the curriculum could easily be a decade-long process, and it would be far from painless.  Being realistic, I suspect that a change like this will only follow from a discussion at the national level. The MAA’s Committee on Undergraduate Programs in Mathematics has already made some recommendations for the role of linear algebra in the curriculum in their 2015 Curriculum Guide.

I generally agree with the CUPM’s remarks, but I think they are too shy in their recommendations. For example, the committee’s Content Recommendation 1 from the Overview says that “Mathematical sciences major programs should include concepts and methods from calculus and linear algebra.” This language seems to put calculus and linear algebra on equal footing (I assume they are listed in alphabetical order), but it doesn’t grapple with the reality that linear algebra is currently under-represented with respect to calculus. I would rephrase their recommendation in stronger terms:

Mathematical sciences major programs should include concepts and methods from calculus and linear algebra. These two subjects should be introduced in parallel, and both should be studied in the first year.

Phrasing it this way emphasizes that a change is needed. Undergraduate programs in the US do not introduce linear algebra early enough and they do not place it beside calculus at the center of the curriculum. Other countries (such as Canada) have done a better job with this, and I hope that we in the US can learn from their example.

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5 Responses to More Linear Algebra, Please

  1. Avatar Brian Winkel, Director SIMIODE says:

    Notions of linear algebra pop up all over the place – we know that – and to not know them or at fail to have exposure to many of the notions/concepts is to be disarmed in problem solving. The notions of linearity are ubiquitous in mathematics because things are quite often linear and if not then we study them in the small where linear is “good enough” to get us through. Sometimes one can use a subject to introduce a linear algebra concept, like eigenvalues and eigenvectors when solving a system of linear ordinary differential equations so there is a give back to linear algebra at times as well as a take.

  2. Avatar jrkrideau says:

    I am not a mathematician but by the time I finished Gr. 13 —precursor to OAC (silly name) and finished first year at Queen’s I’d had two full-year courses in Calculus and one (GR. 13) course in linear algebra and matrices at high school and a full two -term course in both subjects at university. Normal for anyone with a science bent.

  3. Avatar Aram says:

    I totally agree with the general point of this article. One word of caution is that mathematicians sometimes teach linear algebra in excessively formal ways, e.g. doing everything over some field of characteristic zero but not teaching about e.g. Pagerank, as you mentioned. So hopefully linear algebra is taught to general audiences both earlier and in a more applied way.

  4. Avatar Tim says:

    As a minor note… Just because students arrive and either don’t demonstrate proficiency or claim that they’ve never seen complex numbers doesn’t mean that they’ve never seen them.

    Students forget stuff–even stuff that they demonstrated good proficiency with only a few months ago. They’ll claim to have never seen it, even if you’re the one that taught it to them.

  5. Avatar Jacob Koczwara says:

    I agree completely. While I went to an IB high school and took HL math so I was experienced with matrices, vectors, and imaginary numbers i.e. Euler’s form and imaginary algebra my linear algebra skills weren’t as good as my calculus skills. As a math major in college I feel like my education has been more heavily weighed towards calculus. I am a graduating senior this year and I have taken Calculus 1-3, Differential equations and advanced calculus 1 and 2. Whereas to this day I have only taken 1 Linear Algebra course and a Modern algebra course. I feel like this hasn’t prepared me as much as I wish it had for the field of jobs I want to go into. As a future data scientist I know Linear Algebra will factor in much more heavily than Calculus will. To be honest I wish we had had more mandatory Linear Algebra courses. While I feel prepared and am not struggling in my higher level courses I feel like some of my other classmates are. Not everyone has the strong mathematics background and so I feel like the rest of my class is struggling somewhat. Even in Advanced calculus sometimes I feel like I’m one of the few people who fully grasp the material and that the rest of my class is just nodding along in agreement. I feel like extra linear algebra courses would help in this effect and it would definitely help with taking Modern Algebra because while the courses isn’t extremely difficult it would be a lot easier if we were taught more Linear Algebra and if the prerequisite courses were more heavily focused. So in that respect we are in agreement that Linear Algebra should be more heavily focused in undergraduate mathematics degree programs in universities in the United States.

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