Math 2135 (Fall 2020): Linear Algebra for Math Majors

Welcome to Linear Algebra for Math Majors! This is a rigorous, proof-based linear algebra class. The difference between this class and Linear Algebra for Non-Majors is that we will cover many topics in greater depth, and from a more abstract perspective. There will be a correspondingly smaller emphasis on computation in this class, and greater expectations for proof-writing and abstraction.

Contact information

You may also contact me anonymously.

Office hours

My office is Room 204 in the Math Department. My office hours sometimes change, so I maintain a calendar showing the times I will be available. I am often in my office outside of those hours, and I'll be happy to answer questions if you drop by outside of office hours, provided I am not busy with something else. I am also happy to make an appointment if my office hours are not convenient for you.

Syllabus

The following are the main topics of this course. Depending on time constraints, we may or may not cover some of the ones at the end. If we have extra time, we may cover some topics that are not listed here at the end of the course.

1. Vectors and matrices with real coefficients
• linear combination, span, matrix-vector multiplication, matrix-matrix multiplication
2. Vector spaces
• definition, linear combinations, span, linear independence, subspaces
3. Bases
• dimension, coordinates with respect to a basis
4. Linear transformations
• matrix representation with respect to a basis, change of coordinates, linear transformations as a vector space, dual space, matrix multiplication, invertibility, rank
5. Solving systems of linear equations
• reduced row echelon form, invertibility, particular and general solutions of linear systems, bases of image and kernel
6. Determinants
• row and column expansion, multilinearity, relation to invertibility, Cramer's rule
7. Eigenvectors and eigenvalues
• characteristic polynomial, solving for eigenvalues, solving for eigenvectors, diagonalization, existence of eigenvectors, exponentiating matrices
8. Inner products
• length and angle, Gram–Schmidt process
9. Abstract constructions of vector spaces
• products, quotients, spaces of linear transformations, dual, tensor products, symmetric powers, alternating powers

Textbook

The following is the main textbook for this course:

Sergei Treil. Linear algebra done wrong.

We will follow this book closely for at least the first half of the course (Chapter 1 through 4), but we will not always use exactly the same notation in class as the book uses!

The title of Linear algebra done wrong is a response to the title of another linear algebra textbook:

Sheldon Axler. Linear algebra done right.

We will use both Linear algebra done right and Linear algebra done wrong, but the origanization of the course will follow the latter.

Here are a few other linear algebra textbooks that I like:

Hans Samelson. An introduction to linear algebra.

Paul Halmos. Finite dimensional vector spaces.

There are quite a few linear algebra textbooks to be found at the library or by a Google search. The notation will vary, but all cover the same fundamental topics. Please look around and find a reference that fits your learning style.

There are a number of online videos, and some complete courses, about linear algebra. I will assign a few from the following series:

These videos are not a substitute for our lectures and textbooks, but they do give useful visual intuition for a lot of what we are doing. You may also enjoy some of 3blue1brown's other mathematical videos.

Prerequisites

In this class, you will need to be comfortable reading and writing proofs, and using the language of sets. Here is a more specific list of topics:

Logic: be able to read and write direct proofs, proofs by contradiction, and proofs by induction

Set theory: know the definitions of set membership, union, intersection, difference, product, subset, powerset and be able to use them in mathematical reasoning

These topics are all covered in Math 2001 (Discrete Mathematics) and Math 2002 (Number Systems). I will waive the prerequisite for students that can demonstrate their familiarity with these topics.

Course goals

We will learn quite a few definitions and theorems, and a handful of algorithms in this class. The following list is meant to give an idea of the different levels of mastery you can achieve with them. The letters should also give a rough idea of how those levels of mastery will correspond to a final grade.

1. Reproduce the statement of the definition or theorem. Execute the algorithm.
2. Recognize examples and nonexamples of the definition. Recognize settings where the theorem applies or doesn't apply. Identify opporunities to use the algorithm.
3. Write short proofs using the definition. Use multiple ideas in the same problem. Understand how the algorithm works. Transform problems into forms suitable for application of the algorithm.
4. Write long proofs using multiple definitions, theorems, and algorithms. Know and understand the proof of the theorem. Explain how the algorithm works. Apply the algorithm in novel situations. Adapt the algorithm to solve different problems.

Homework

Homework will be assigned with every lecture. It will be collected in the subsequent lecture, and returned with comments, but daily assignments will not be graded. The only graded work in this class are the approximately bi-weekly exams and problem sets.

The goal of homework problems is to provide practice with the concepts we encounter in class. The point of submitting homework problems is to get feedback on your work. This means you do not have to submit every problem that has been assigned: only write up and submit the problems you want feedback on! You can and should skip problems you find too easy or too difficult to be useful. (Please let me know if you are finding our homework problems too easy to too difficult.)

You should plan to spend approximately 3 hours between lectures on the class. Not all of this should be spent on homework problems: I recommend spending only 1 to 1.5 hours on homework problems, even if this means you do not finish all of them. In the remaining time, you should do the reading assignment for the next lecture and review past lectures and readings.

Expectations

This course has high expectations. You should plan to spend 9 hours per week on this class, not including lecture. It will also be necessary to supply independent motivation as not all of the work you need to do for this class will be collected, or even assigned. It is also essential to recognize early when you are struggling with a concept and discuss it with me.

Above all, you must engage actively with the material as we learn it. If you are studying actively, you will have questions. Use this principle to measure whether you are actively engaged.

The first thing to do when joining this course is to make sure these expectations align with your goals.

I encourage you to consult outside sources, use the internet, and collaborate with your peers. However, there are important rules to ensure that you use these opportunities in an academically honest way.

2. Plagiarism will be dealt with harshly. Deliberate plagiarism will be reported to the Honor Code Office.

To avoid plagiarism, you should always cite all resources you consult, whether they are textbooks, tutors, websites, classmates, or any other form of assistance. Using others' words verbatim, without attribution, is absolutely forbidden, but so is using others' words with small modifications. The ideal way to use a source is to study it, understand it, put it away, use your own words to express your newfound understanding, and then cite the source as an inspiration for your work.

Questions

Do you have a question or comment about the course? The answer might be in the course policies, on this page. If your question isn't answered in the course policies, please send me an email. Or, if you prefer, you may send me a comment anonymously.

Your grade will be the average of your scores on approximately 3 in-class exams, approximately 4 problem sets, and the final exam. Each exam and each problem set will count equally, and the final exam will count as the equivalent of 2 exams. You should plan for exams to occur on, and problem sets to be due on, Fridays.

The scoring on these assessments will be based on the goals listed above. Notably, your score will not be a simple sum of point values from each problem, but will instead be my overall assessment of the degree to which you have achieved the course's goals on the relevant topics.

Missed exams and problem sets will be addressed using the revision policy. No grades will be dropped, apart from those replaced by revisions.

Revisions

At the end of the semester, I would like for your final grade to reflect your mastery of the course material. Exams and problem sets do not always measure this optimally, so you will be allowed to revise your scores by the following process: 1) decide which score you wish to revise; 2) identify the topics that were assessed (for example, from the course outline) and put these in a list to be handed in with your revision (you may want to clear your list with me before going on to the next step); 3) find or devise a list of problems that you can use to demonstrate your mastery of those topics (again, you may want to discuss these with me); 4) solve those problems and submit your solutions to me. I will assign a replacement grade based on your submission.

As a practical matter, I insist that your revisions be submitted within one week of the return date of the original assignment. This is meant to prevent an influx of revisions at the end of the semester, when I will not have time to look at all of them, and to ensure that you receive the benefit of doing a revision in time for it to be relevant to the rest of the course.

In your revision, be sure to include all of the following:

1. a cover page indicating which assessment you wish to revise, a list of the topics addressed in that assessment, and an index that I can use to find which of the problems in your revision cover those topics; making this list will require studying the original exam or problem set;
2. a list of problems and solutions covering the topics named above; you must find or create these problems yourself — new solutions of the problems from the original exam or problem set will not be accepted!.