# Make a matrix algebra course (1st university year) more "project-based"

Among other courses, I'm teaching a (basic) matrix algebra course for 1st year university students (they are studying Economics, and the cursus leads them to management, finance, or econometrics in 3rd univ year).

As it's quite common in France, these notions are taught in a rather abstract way, we teach them: matrix operations, determinants, rank, matrix equivalence, row echelon form, application to linear systems, optimum of functions, etc. (but not eigenvalues/vectors, this is done the next year).

It's a ~ 40-hour course in a semester, I have ~ 40 students per group, and I'm doing nearly 1/3 theory, 2/3 training exercises during these 40 hours.

The problem: even if 1) it works quite well after a few years teaching this course 2) noone complains 3) these notions are important for them for their following years (econometrics, they will also learn R, etc.), I feel the course is a bit "too abstract" / not enough connected to applications and not pro-active enough for them.

Idea: I would like to test something like that:

• Do the usual 40 hours in ~ 34 hours (I can reduce the number of exercices of 20% more or less)

• In the 6 remaining hours, I'd like to test something more active for them, more project based, example:

• First 1.5hr sequence: I present the project briefly, and give "topics", they start working in ~8 groups of 5 students
• Homework: they do some research on their "project"
• Second 1.5hr sequence: they continue their research, I help them / answer questions, give them directions if needed, they bring books from university library + their laptop to find additional resources
• Third 1.5hr sequence: half of the groups do a presentation of their "project"
• Fourth 1.5hr sequence: the other groups do the same

Questions:

• Have you done such "project based" courses for university students? Any feedback?
• The biggest question: what kind of math project could they work on? (using some matrix/linear algebra). I have a few ideas (see here for example, I used some of them for 3rd year students), but nearly all of these applications are too complex for students studying matrices for the first year.

• What kind of setup would you use? Provided they are econ students, I was thinking about taking business-related examples.

• Check Philip Klein’s coding the matrix and Time Chartier’s Life is Linear Mar 23, 2018 at 4:08
• If you know programming, it works perfectly. Back in my early undergraduate days I decided to take a break from theoretical mathematics to take a more applied linear algebra course, which used Numpy extensively to give hands-on experience working with matrices. It was extremely helpful, even for my theoretical work and understanding! Mar 27, 2018 at 18:55

I've tried something of the same form, engineering students.

Line of projects were both applied and theoretical:

• Simple linear programming problems

• Simple population models

• Discovering the link between determinant and volume

• Optimization problems presented in game form (which could be solved by setting a problem and solving linear systems)

• Problems on Graphs

My experience is that time runs out very fast. I underestimated completely the time required for my students to understand that some problems could be formalized with linear algebra. Mostly because they did not know how to model a problem altogether. So that they were drawn towards "finding solution" more than "building a good model". They needed a lot of directions on how to build a model and understanding even the basic issues in the model.

If you have many groups and you're alone in interacting, you will have to jump from one to the other. Asking questions of 8 groups? No one of the groups understand much about questions of the other therefore gets distracted and may become noisy. Organize this session VERY carefully.

On the positive my students had a lot of fun and they participated in this very actively. It was a good experience, it took me a lot of time to build up decent models, maybe they haven't learned as much as I would have hoped out of it, but overall going back I would do it again (considering also what I gained from it in terms of teaching experience).

Project example.

Car Rental Company. Rents car in three cities A, B, and C. Each person can return the car in the same or in a different city. We have tables on statistical habits of people: 70% of renters in city A returns in city A, 20% in B, 10% in C and so on.

We start with 100 cars and a certain distribution of them along A, B, C. What will be the distribution of cars after 1 month, 6 months, 1 year, in the long run? Is there an initial distribution of cars granting equilibrium? What initial distribution of cars should I have to have them equally distributed after 3 months? How do I take into account cars being stolen at a different rate in the three cities? What about new cars being bought?

(and so on)

Addressing the point about the course not being sufficiently connected to applications, you might be interested in Boyd & Vandenberghe's recent Introduction to Applied Linear Algebra: Vectors, Matrices, and Least Squares (VMLS). The book is available for free online, with permission from the publisher. It includes numerous applications, some of them from business/economics. Some of the applications in the first half of the book, which focuses on vectors and matrices, are:

• Leontief input-output models
• portfolio sector exposures
• supply chain dynamics
• conversion from purchase quantity to purchase currency unit (dollar, euro, etc.) matrix

Eigenvalues and eigenvectors are not dealt with in VMLS, but you say that these are not covered until the course in the second year, so that shouldn't be a problem.

You don't mention least squares specifically, but if you do cover them or decide to introduce them, VMLS devotes the entire third part of the book (chapters 12-19) to this topic, starting with basic least squares and ending with constrained nonlinear least squares. Some applications from business/economics in these chapters include:

• predicting return of an asset from the return of the whole market
• time series trends and predictions
• regression models
• fraud detection
• marketing demographic classification
• product demand shaping
• minimum cost trading to achieve target sector exposures
• portfolio optimization
• finding equilibrium prices
• internal rate of return

If you want to include programming in the course, there are companions available for using Julia and Python.

(If the authors sound familiar by any chance, they also wrote the more advanced book Convex Optimization.)

Could take a look at some texts like this: https://www.amazon.com/Matrix-Algebra-Business-Economics-Shayle/dp/047176941X

I like the idea of a a special project business case: optimizing a factory, refinery, catalog business, or insurance agency. Because I think economists are weak on basic behavior of firms. But I worry, you don't really have time for all that.

But the second link has some very simple business problems (not even projects, just normal word problems). Could consider to intersperse some of these into the course rather than an end project. Also, the level of difficulty seems similar to what you are doing.

Here are projects I have written and used. I assign these to student groups of 3-6 students, mostly Engineering majors, mostly sophomores. In class time is about 30 minutes per project to introduce the set up; then they work out of class for about 2 weeks per project (and come to office hours frequently during that time).

I want to make another one on Fourier transforms.

If anyone would like to use these, send me an e-mail and I'll send you LaTeX source and auxilliary files. Please give me credit. At some point, I'll get around to doing a formal Creative Commons release of these.