a few facts
Basic Info
- Instructor: Dave Feldman
- Email: You can figure it out
- Office: Second floor, Turrets Annex
- Pronouns: he/him/his
- Help Sessions: TBA, Dining Hall
- Office Hours: By appointment
- Tutors: None
i guess basically i want you to learn linear algebra and also some differential equations
Course Goals
- I want you to learn what differential equations are and how to think about them: how they are used to model a range of phenomena and how to interpret their solutions.
- I want you to gain a firm foundation in the basic concepts of elementary linear algebra, including: vectors, matrices, inverses, determinants, vector spaces, linear independence, sub-spaces, basic, dimension, rank, eigenvalues and eigenvectors.
- I want you to gain an introduction to some of the basic analytic techniques used to analyze linear differential equations, including systems of linear equations.
- I want you to gain mathematical confidence, appreciation, and "maturity". As part of this, I want you to continue to work toward developing a careful, systematic, and effective problem-solving style.
- I want you to have fun and learn a lot.
so, umm, what is linear algebra, anyway?
What is Linear Algebra?
Wikipedia's entry on linear algebra has a pretty good summary:
Linear algebra is a branch of mathematics concerned with the study of vectors, with families of vectors called vector spaces or linear spaces, and with functions that input one vector and output another, according to certain rules. These functions are called linear maps (or linear transformations or linear operators) and are often represented by matrices. Linear algebra is central to modern mathematics and its applications. An elementary application of linear algebra is to the solution of a system of linear equations in several unknowns. More advanced applications are ubiquitous... . It has extensive applications in engineering, physics, natural sciences and the social sciences. [Linear algebra is also important to study because] nonlinear mathematical models can often be approximated by linear ones.
Linear algebra is a very standard topic in the college mathematics curriculum. It is usually taken by students in their sophomore year. Linear algebra is generally required for math, physics, engineering, statistics, and economics majors. While linear algebra is a standard subject, there are considerable differences among linear algebra courses and textbooks. Roughly speaking, there are three somewhat overlapping roles that linear algebra plays in the math curriculum:
- Linear algebra teaches a set of techniques for solving problems. These classes often emphasize applications.
- Linear algebra serves as a bridge toward more formal or abstract mathematics.
- Linear algebra can be an entry point for numerical mathematics. This includes: Using a computer to do mathematics and solve problems and analyzing the run-time and convergence properties of algorithms, writing original code.
This course is mostly of type 1 but will contain elements of 2. We will do only a little bit with computation in this class.
This course is a combined linear algebra and differential equations course. While most math programs have separate linear algebra and differential equations courses, combined courses such as this one are not uncommon. Such combined courses are often part of engineering math sequences. (Sometimes such courses are called linear systems.)
Linear algebra and differential equations are pillars of modern
applied mathematics. I think it makes sense to do an introduction
to them at the same time. This course emphasizes conceptual and
analytic (pencil-and-paper) understanding of linear algebra and
differential equations. My differential equations course focuses on
computational methods and mathematical modeling. The two courses
are designed to complement each other. Please note that a course in
python (or the equivalent) is a pre-req for differential equations
at COA.
these are cautionary notes for me as well as you
Some Warnings
- I have taught linear algebra and differential equations many different time, but only once before as a combined class. In order to make this class work---really see the connections between linear algebra and differential equations---we're going to need to cover a lot of material. We can do it, but maintaining a brisk pace will be necessary.
- Most math classes the feel of the class is fairly homogeneous. Some topics are a bit more difficult than others, but the class doesn't have any sudden shifts. Not this class. We'll be covering very different topics that will feel very different.
- Linear algebra potentially has some trap doors: things can suddenly shift from easy to very hard without you realizing it. Of course I'm going to try to not let this happen.
- For most of you this will be your first "sophomore level" math class, so it's a notch more advanced/abstract from that which you're used to. So you might feel a little unsettled. This is normal, and that sense of unsettledness might not fully go away until your second or third math class at this level.
- My experience as a student was that linear algebra seemed very easy. So I didn't do much work in the class. But then it got harder quite quickly and I was lost and in a lot of trouble. As always, it's important to stay caught up, because you can get left behind pretty quickly in a course like this.
- Also, when I took linear algebra I remember thinking that the material was kinda silly and that I would never use it. However, I ended up using linear algebra in almost every math and physics class I took, and I used it in my research all the time---much more than I use calculus. Some of the applications of linear algebra might not be apparent immediately, but please believe me when I say that it is a very useful topic to know.
- Introductory linear algebra is at times an odd mixture of tedious and straightforward calculations and austere and potentially difficult abstractions. It has a very different feel to it than calculus. I think most of you will enjoy it.
nuts and bolts and further thoughts on the class
About this Course
- There will be weekly assignments that you'll turn in at the end of the day Friday. (I come to campus Saturday late morning and retrieve assignments.)
- We'll do some sort of a low-stakes in-class (but probably open notes) midterm during the middle of the term. I don't anticipate there being a final exam.
- I do not expect all of the assignments to be easy. You'll want to take advantage of help sessions and I also suggest working collaboratively. There is no TA for this class, so we'll have to work to find times for help sessions that work for everyone, including non-COA students. I'm open to experimenting with online help sessions.
- An added challenge is that I will be on campus less than usual this term. I'll be traveling some due to my work with the Santa Fe Institute, and I also have some significant dental scheduled for week two. We
- Your evaluation will be based roughly as follows:
Homework 90% In-class midterm 10% - If winter weather conditions make it unsafe for you to come to campus, do not come to class, even if classes have not been officially canceled.
collaboration vs copying, and other details
Homework
- Any code that you submit should be uploaded to our shared google drive. Please don't email me code. Thanks.
- On the one hand, coding is a solitary activity. You need to learn how to write code on your own. On the other hand, coding is a collaborative activity; one often discusses ideas and strategies with friends and colleagues. Additionally, one sometimes grabs snippets of code from the web or even an entire program.
- Try to do as much coding on your own as possible. Especially in the first half of the course, write from scratch, or by starting with code we work on together in class.
- Never hesitate to talk about coding ideas or challenges or strategies with others. It's ok to get help debugging, but resist the urge to get help immediately every time. Debugging is an important skill. As the course progresses, you'll get better at it.
- If you share code with others, that's ok, but I'd encourage you to not do this often. If you do look at others' code and/or look at code on the web, be sure to cite your sources via comments in your code. If at all possible, make sure you understand the code you use. Sometimes that's not possible, in which case you should note as much via a comment in your code.