Linear Algebra

This is an archive of the first 6 weeks of the linear algebra course I taught at Emory University in Spring 2021. I cross-posted all class pages and notes both here and on the Canvas page, but students much preferred Canvas and it got overwhelming to keep two websites up to date so I gave up posting here after the first half of the semester. I hope the incomplete information presented here is still somewhat helpful.

Please scroll all the way down for general information about the course, or click here for a PDF file with the course syllabus and policies.

Week 6 Day 3 (Lab):

Here's a link to the worksheet we will use in class:

Week 6 Day 2

In this class session, we will begin Chapter 3: Determinants and Diagonalization.

Link to notes

Week 6 Day 1

In this class session, we will continue talking about (linear) transformations. This includes material from sections 2.2, 2.4, and 2.6.

Link to notes.

Upcoming deadlines

Hey everyone,

There has been, understandably, some confusion about upcoming deadlines, so I wanted to send out an email to clarify what's due and when.

In the next week, you will be working on 3 things for this class:

- HW 5 (PDF available here and due on Gradescope.)

- Chapter 2 Summary (PDF available here and due on Gradescope.)

- Exam Wrapper. (Canvas survey, accessible via this link.)

The official deadline for all three things is March 12th at 1 pm. My reasoning for keeping all three things due at the same time is so you won't have to remember three different dates/times for this class along with remembering deadlines for other classes you may be taking.

However, if having 3 things due at the same time causes you stress, here are 3 suggested deadlines for each of the above items.

- HW 5 (Friday, March 12th at 1 pm on Gradescope.)

- Chapter Summary (Monday, March 8th at 1 pm on Gradescope)

- Exam wrapper (Whenever you look at your exam grades, please spend a few minutes on writing this reflection.)

The suggested deadlines are just for you and I won't penalize you if you don't meet them.

Please feel free to email me if you have any questions, comments or worries ;)



Week 5 Day 3

Exam 1 will be held in class on Friday at our usual time. It covers sections 1.1-1.3 and 2.1-2.5.

Exam instructions!

Hi everyone,

As promised, here's my email with a long list of exam instructions.

1. As you all hopefully know by now, our first midterm exam will be held in-person during class time on Friday, February 26th.

2. Please ensure that you arrive on time, and if possible, a little early, so we can all be seated and begin exactly at 9:40 AM. Please note that the exam will start promptly at 9:40 AM and if you're late, that will cut into your own exam time.

3. As you walk in, please turn your phone off, and put your backpacks and all other belongings (other than stationery) in the front under the blackboards. Please take care to social distance from each other the entire time. Please note that turning your phone off is different from putting it on silent or vibrate. This is to avoid a situation in which someone's phone makes any noise in their backpack during the exam.

4. Exam booklets will be on the bench that I usually leave lab materials on. Please grab one as you walk in, bring it to your desk, write your name on top, and read the instructions on the cover page. Please do not open the booklet until instructed to do so!

5. You're only allowed pens/pencils, erasers, and/or other stationery on your desks. No scratch paper, calculators, or other electronic devices are allowed. If you need scratch paper during the test, I'll have some at the front of the classroom. Please raise your hand, wait for me to acknowledge your request, then get up to get scratch paper.

6. The exam is 50 minutes long, it is written so that if you know the material well, it should take you less than 50 minutes to finish it. However, there isn't time to waste built in, so if you get stuck at a problem, move on and come back to it at the end. Don't spend too much time staring at a problem until you're done with the rest of the exam.

7. Once you're done with the exam (or when I call time), please leave your exam booklets at your desk and exit the classroom while maintaining a minimum of 6 feet of distance from everyone else. This might mean that you have to wait for other people to get their backpacks before you can get to yours. Please be patient. All rules of a pandemic classroom still apply during an exam.

8. Most importantly, good luck! Remember to get some sleep and eat something light before the exam. In particular, if you usually eat breakfast in the morning please plan ahead and try not to skip it on exam day. (If you usually don't eat breakfast, Friday might not be the best day to start a new habit either.)

Please feel free to email me with any questions or concerns at any time.

Happy Studying!


Week 5 Day 2

In this class session, we will finish covering Section 2.5, review some material for the exam on Friday, and begin talking about transformations (Sec 2.2).

Class notes are now available here.

Zoom recording is available on Canvas.

Week 5 Day 1

In this class session, we will finish talking about Section 2.4: Matrix Inverses and start talking about Section 2.5: Elementary Matrices.

Class Notes are now available here.

Week 4 Day 3 (Lab):

Here's a copy of the worksheet we will use in lab:

Week 4 Day 2 (Rest Day!)

Wednesday, February 17th, is a university-wide rest day! Please take some rest!

Week 4 Day 1:

In this class session, we will go over Section 2.4: Matrix Inverses.

Class Notes are now available here.

Week 3 Day 3 (Lab):

Here's a copy of the worksheet we will use in lab:

Week 3 Day 2:

In this class session, we will work on Section 2.3: Matrix Multiplication.

Before you come to class, you can think about the following problem:

Suppose that A and B are two m x n matrices such that Ax=Bx for EVERY n-vector x. What can you conclude about A and B?

Class Notes are now available here.

Week 3 Day 1

In this class session, we will begin Section 2.2: Matrix-Vector Multiplication.

Before you come to class, you can attempt to solve the following problem:

Write the general solution to the following system in parametric form

x − y − z + 3 w = 2

2 x − y − 3 z + 4 w = 6

x − 2 z + w = 4

Week 2 Day 2

In this class session, we will finish our discussion on Section 1.3, Homogenous Equations, and begin Chapter 2: Matrix Algebra.

Before you come to class:

Consider the following problem: Let A be an n x m matrix of rank r and consider the homogeneous system whose coefficient matrix is A. How many solutions does this system have? How many parameters are in the general solution?

Homework 1 is due on Friday, Feb 5th, on Gradescope at 1 PM.

Homework 2 is now available here. It's due a week from Friday on Gradescope at 1 PM.

Week 2 Day 1

In this class session, we will finish talking about the rank of a matrix, from Section 1.2. We will start Section 1.3: Homogenous Equations.

Recommended problems from Week 1:

If you'd like more practice from the contents of last week, you can look at the following problems from your book:

1.1.14-1.1.15, 1.1.18-1.1.20, 1.2.12, 1.2.16-1.2.22.

Week 1 Day 3: Lab

In this class session, we will go over the following worksheet together. If you'd like to print out a copy and bring it with you, you are welcome to download it here:

Week 1 Day 2:

In this class session, we will cover the rest of Section 1.1, and start 1.2.

Before you come to class:

- Please consider the problem: Describe all solutions to 3x-y+2z=6 in parametric form in two different ways.

- Please fill out the Introductions! survey if you haven't already.

- Please confirm that you can access both Gradescope and Zoom chat! I will ask you to solve a problem in class and send it to me over Zoom chat in real-time.

Please submit by Friday, February 5th at 1 PM EST, over on Gradescope.

Week 1 Day 1:

In this class session, we will go over the syllabus, expectations for attending an in-person class during a pandemic, and Section 1.1 of our textbook.

Before you come to class:

  • Please carefully read over the syllabus file, make note of any questions you have, and come prepared to ask those questions in class.

  • At some point during Week 1, please complete the survey titled: Introductions! During class, I will ask you to tell us about yourself in a few sentences. Filling out the survey might help with coming up with an introduction beforehand :)

  • Here is a famous problem that uses linear algebra for you to think about if you are interested:

    • (The Hundred Fowls Problem.) One hen is worth 5 qian, one rooster 3 qian, and 3 chicks are worth 1 qian. If we bought 100 fowls with 100 qian, how many hens, roosters, and chicks did we buy?

Important note: Please don't feel discouraged if you cannot immediately solve the above problem. I haven't taught you anything yet! This is meant to give you a flavor of the kinds of things we will be talking about on Day 1.

Welcome to Math 221

Hi all,

Welcome to Math 221 - Section 7. I am excited to work with you all this semester.

Please read on for important information pertaining to this course:

  • Here is a link to the syllabus. The syllabus contains a link to the textbook. We will go over the syllabus in class on Day 1, but please read it thoroughly and come prepared with any questions. We will also cover Section 1.1 of the textbook.

  • Note that our first class will meet in-person in the Math and Science Center, W201 on Monday at 11:20 AM - 12:35 PM.

  • This bears reiterating: Even though we are meeting in person, we are still in the middle of a pandemic. What this means is that we cannot get lax about health and safety measures in our classroom. Please cover your nose and mouth with a mask the entire time you are in class, and keep a safe (>6 feet) distance away from everyone, including my desk at the front. When you enter the classroom, please try to sit as far away from the door as possible, so no one has to walk past you to find a seat.

  • Now that that is out of the way: at this link, you can find a beginning of semester survey hosted on Canvas. This is a means for me to get to know you a bit before the semester starts. Please answer the questions in as much detail as possible.

  • Here you can find a Canvas page dedicated to the first class session. I will make a similar page for every session, and it will be your catch-all place to find notes, recordings, other resources, etc.

  • Finally, other than Canvas and Zoom, we will be using Gradescope for submitting homework in this class. All of you should have already gotten an invite to the gradescope course, but please email me if you haven't and I will help set it up.

Please feel free to email me with any questions or concerns! I look forward to seeing you all on Monday!

Why linear algebra?


Click here for a PDF file with the course syllabus and policies.

Course Websites.

  • All of the information presented here (and more) is also available on the Canvas site for this course.)

  • Other than Canvas, we will use Zoom and Gradescope in this class.


We will use the following textbook which is free to download:

The book consists of eleven chapters, we will cover most parts of the first eight chapters.

Miscellaneous Notes: