2024 December 3,

Math 134: Linear Algebra with Applications

Carleton College, Fall 2024, Dr. Joshua R. Davis, , CMC 324, x4095

Introduction

In my opinion, linear algebra is the most important kind of math. We begin with a simple idea — solving systems of linear equations — but it turns out that this idea pervades mathematics. It has numerous practical applications: web search engines, error-correcting communications protocols, cryptography, computer graphics, relativity, quantum mechanics, balancing chemical reactions, structural geology, and much more. Because so many people use linear algebra, it can be seen from many different viewpoints: matrix/tensor algebra, intersecting hyperplanes, abstract vector spaces, etc. So there is a lot to say about the simple idea of solving systems of linear equations.

Math 134 is an introduction to linear algebra. It has the same curricular standing as Carleton's other linear algebra course, Math 232. For example, later courses accept Math 134 and Math 232 equally as prerequisites. The main difference between the courses is that Math 232 has some abstract vector space material, while Math 134 instead has more in-depth applications and computation. In both courses, students do lots of calculations and logical reasoning using algebra and geometry.

Unlike Math 232, which has some calculus prerequisites, Math 134 has no college-level prerequisites. Students are expected to be comfortable with high-school-level algebra. After some tutorials, they are expected to use the computer program Mathematica for large calculations and visualizations. Talk to me if you are concerned about your background.

Responsibilities

The College's accreditation says that a 6-credit course is 150 hours of work. That's about 15 hours per week or 5 hours per class meeting. Those 5 hours break down into about 1 hour for class itself and 4 hours for homework, reading, studying, etc. If you find yourself spending less time and struggling, then talk to me. If you find yourself spending much more time, then talk to me.

Class Participation

Our class meets in CMC 210 during period 3A (MonWed 11:10-12:20, Fri 12:00-1:00). You are expected to attend every class meeting, take notes on paper or a tablet, participate in discussion and group work, and ask and answer questions. You can make up for a deficiency in class participation by talking with me in office hours.

Laptops, phones, photos, and recordings are prohibited (except by special arrangement). Why? This course's material can't be typed easily on a keyboard. Using a laptop or phone measurably distracts the students around you. Photographing a chalkboard is not as educational as taking notes. I want our class to be a safe space, where students do not feel that they're on stage.

It is important that our course be welcoming to all students, regardless of their identities, backgrounds, and experiences. We all sometimes say and do things that make life worse for others, and we should all strive not to. Please let me know if the class feels hostile to you, because of something that I or someone else has done.

Homework

Although class meetings may seem like the core of the course, homework assignments are actually where you learn the material. As far as your grade is concerned, homework serves primarily as the first step in studying for exams.

On homework, you are encouraged to figure out the problems with other students. However, you should always write up your solutions individually, in your own words. You may not copy someone else's work or allow them to copy yours. You may use Mathematica or a calculator to expedite calculations, if you mention so in your solution. You are not permitted to obtain solutions from web sites (other than this one), artificial intelligence products, etc. Presenting someone else's work as your own is a violation of Carleton's Academic Integrity standards.

Writing is not just for English and history majors. Communication skills are essential to every academic discipline, and they are highly prized by employers. In this course, your written work is evaluated both for correctness and for presentation. Compose your solutions as if the intended audience is your fellow students. By doing so, you show enough detail that the reader can ascertain whether you yourself understand the material.

Homework is usually due two class meetings after it was assigned. This schedule tries to keep you on track, so that work doesn't pile up. But we all have bad weeks, where we can't get everything done, right? If you need to submit an assignment late, then write "LATE" at the top of the front page and submit it as soon as you can. If the grader hasn't graded the assignment yet, then they can grade your paper with the others for full credit. If the grader has graded the assignment already, then you might not get credit. There are limitations to how much delay and complication a grader can handle.

When handing in an assignment, please staple your pages into a single packet, in the correct order. Is there a stapler in the classroom? Often not, so staple ahead of time. Is a paper clip just as good? Sorry, no.

Depending on time constraints in any given week, perhaps not all of your homework is graded. In order to ensure full credit, do all of the assigned problems.

Exams

We have three exams. Exam A is given in class about one third of the way through the course. Exam B is given in class about two thirds of the way through the course. It is cumulative but focused on recent material. Exam C is scheduled for the official final exam period: Monday November 25, 8:30AM-11:00AM. It is cumulative but focused on recent material. Self-scheduling is not permitted for Exam C.

Grading

For better or worse, we are required to measure your learning using grades. Your numerical grade is based on the responsibilities above: participation 5%, homework 15%, Exam A 25%, Exam B 25%, Exam C 30%.

Numerical grades are converted to letter grades only at the end of the term. Grades are not curved, so students are not in competition with each other. Grades are also not based on predetermined percentages (90%, 80%, 70%, etc.) or explicit specifications, because college-level problems are difficult to tune so precisely. (I could accidentally write a difficult exam and wreck everyone's percentages.) Rather, I assign grades by comparing the students to the course goals. For example:

• A student, who earns an A, is not perfect, but they can speak fluently about nearly all of the course content. They exceed expectations.
• A student, who earns a B, learns most of the big concepts pretty well, but they are missing some concepts. They lack the depth of understanding needed to appreciate subtleties.
• A student, who earns a C, gets some of the material but has significant gaps in their understanding of major concepts.

A disadvantage of this system is that, at any given time, you cannot compute your current letter grade. Talk to me, if you are concerned and want an estimate of your current grade.

Resources

I want all of my students to work hard and learn a lot. I try to give them all of the resources that they need. For starters, here are some texts.

• Linear Algebra And Its Applications, 6th edition, by Lay, Lay, and McDonald is our official textbook. Many of our homework problems come from the book. The best way to study for exams is to do additional problems.
• Discussion Problems is a growing list of problems that we discuss throughout the term in class.
• Installing Mathematica on Your Own Computer is free, and it lets you work on your own computer. Alternatively, you can work in the labs (CMC 304 or 102) at most times of day.
• introduction.nb is an elementary Mathematica tutorial. (Right-click or Control-click to save this Mathematica notebook to your computer. Then double-click to open it in Mathematica.)
• memory.nb is a second Mathematica tutorial.

Here are our exams from this term, with quartiles (75th, 50th, 25th percentiles).

• Exam A, Solutions (46/52, 40/52, 36/52)
• Exam B, Solutions, Practice (51/53, 45/53, 41/53)
• Exam C, Solutions, Exam C Practice (64/68, 58/68, 53/68)

Remember that I encourage you to solve problems with classmates (even if the work that you submit must be your own). E-mail me, if you want help in arranging a study partner. In your e-mail, you might summarize your preferences: getting started early versus waiting until the deadline, working in the evening versus late at night, which classmates you can't work with, etc.

I hold several office hours per week. See below for the tentative schedule. No appointment is needed; just drop in! If you cannot attend office hours, then consult my weekly schedule and e-mail me, listing several times at which we could meet. Our course prefect, Owen (e-mail formano), also holds prefecting sessions. Again, just drop in or e-mail for a special appointment.

The tutors in the Math Skills Center on the second floor of CMC can also help.

If a health condition or other personal matter affects your participation in class, homework, exams, etc., then please let me know as soon as possible. Depending on the situation, we might want to confer with Accessibility Resources, Assistive Technology, Student Health and Counseling, or Sexual Misconduct Prevention and Response. When you ask me to help, I do my best to help. :)

Schedule

To help you decode the schedule, here is an example. Monday September 16 is Day 1 of the course. On that day, we discuss systems of linear equations. You have homework called "Day 1". Attempt it immediately, but hand it in at the start of class on Day 3 (which is Friday September 20, as the schedule shows). Read Section 1.1, if you like, to get another treatment of the material. There is also a Mathematica notebook, linearSystems.nb, which you can peruse.