Alexander Paulin 

Department of Mathematics
796 Evans Hall
University of California, Berkeley

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Linear Algebra and Differential Equations (Math 54) Spring 2022

Except for the first two weeks of class, lectures will be held in-person at the following times and locations:

LEC 001: 10am-11am MWF, Dwinelle 155

LEC 002: 11am-12pm MWF, Wheeler Auditorium

Both lectures will be indentical and you are welcome to attend either, although it would be best to attend your offical section for the first few weeks.

Attendance is not mandatory (although, I'd strongly advise that you go). In-person lectures will not be recorded, although detailed asynchronous videos will be provided below.

For the first two weeks, all lectures will be held remotely at the above times at the following Zoom Link:

Every Monday, Wednesday and Friday you will have a one hour discussion session led by a GSI (graduate student instructor). It's in these sessions that you'll really explore the material as a group (around 25 students in each section). In discussion section you'll work together to solve challenging problems. This is where you'll really learn how to master the material. Here's some advice about how to get the most out of discussion section:

  • Learning Mathematics can be really challenging. As with almost anything in life, making it a social experience will make the process much more rewarding and productive. View discussion section as a way to both learn the material and potentially find people to work with outside of class.
  • Don't be nervous about asking questions (or answering them) in group discussions. Even if you're not correct, lots of other people will definitely be thinking the same thing and it'll be a learning experience for the group. There are often many ways to approach a problem and hearing other peoples' perspectives can dramatically deepen your understanding.
  • Your GSI will frequently split the main group into smaller groups where you'll work on problems more collaboratively. Be an active participant in these. Even if you're not completely confident in your approach, still share it.

For a discussion section schedule go to the folllowing links: Section 001 and Section 002.

Attendance at discussion section is not mandatory (although I'd strongly advise you to go).

You can only attend the section you are officially enrolled in. If you have enrollment questions contact:

For the first two weeks, all discussion section will be held remotely. You will be contacted by your GSI with more details

Here is a link to GSI contact information.

In addition to lectures and discussion section, you will have access to daily office hours. Office hours are you're chance to talk to me or a GSI. They are a really good way to get to know your instructors better and get help with any aspect of the course. These are also a space for you to work in small study groups at your own pace if you'd like.

In office hours you can talk about any aspect of the course (and beyond). If you've spent time on a homework problem and have stalled, come to office hours; If you're unsure about your academic trajectory, come to office hours; If you want to learn what mathematics research is about, come to office hours; If you're struggling with the course and don't know what to do, come to office hours; If you think you might need a reference from a professor in the future, come to office hours; If you just want a chat, come to office hours.

For the first two weeks, all office hours will be held remotely at

After the first two weeks of class, all office hours will be in-person.

Here is a link to the complete office hour schedule.

Linear Algebra and Differential Equations (UC Berkeley Custom Edition), ISBN: 9780137438457.

This is a combination of two separate textbooks. These textbooks are

Linear Algebra and its Applications, Lay-Lay-Macdonald, 5th Edition.

Fundamentals of Differential Equations, Nagle-Saff-Snider, 9th Edition

I advise you against buying these textbooks individually. They contain far more material than will be needed and will be substantially more expensive.

Homework assignments are due on Gradescope each Tuesday at Midnight according to the schedule posted below. Here are instructions for how to upload your work. Your two lowest homework scores will be dropped.

Assignments will be graded on completion. Your two lowest scores will be dropped. You may discuss the homework problems with your classmates, but you must write your solutions on your own. Make use of discussion sections, office hours, study groups, etc. if you need assistance, but in the end, you should still write up your own solutions.

Lectures are where new concepts are introduced, but homework is where most of the real learning happens. It's where you internalize the abstract ideas and discover for yourself how they can be used to solve problems. From my experience the main distinction between those who succeed in this course and those who don't, is how they treat the homework. Here's some advice about how to approach it:

  • Be organized. Don't leave things for the last moment. (You'll struggle to complete the homework assignment if you start on the night before it is due.) Work consistently in small installments. Some homework can take 6 or more hours to complete properly.
  • As you progress through the homework you'll notice it increasing in difficulty. The more difficult problems are often the most important, giving you the opportunity to really master the material. If you're struggling with a challenging problem you should spend at least 30 minutes on your own, thinking about it in depth. Even if you fail to make a breakthrough, this is still more worthwhile than giving up after a couple of minutes and talking to a peer or consulting an answer scheme. In future courses you'll be solving problems that require days (or weeks) of dedicated thought. Now is the time to hone this skill.
  • If you've spent serious time exploring a problem and are still struggling, then is the time to seek help: Speak to your peers, come to my or a GSIs office hours, go to the SLC and talk to a tutor. The last thing you should do is look at a solution manual. Actively speaking to someone about a problem is much better than passively reading a solution.

Here is a link to the homework schedule and assignments.

Quizzes will take place during Wednesday discussion section according to the following schedule. They will last about 15 minutes and be closely related to the homework submitted the day before. If you've done the homework carefully, the quiz will be strightforward. Your two lowest scores will be dropped from your grade. Here is the quiz schedule:

1 2/2
2 2/23
3 3/2
4 3/9
5 3/16
6 4/13
7 4/20
8 4/27

There will be no make-up quizzes, unless there are exceptional circumstances.

There will be two midterms and a final exam. All exams will be in-person without exception.

Because of the grading scheme, you can miss one midterm, for whatever reason, without penalty. On the other hand, missing both midterms will seriously harm your grade and make it much more difficult to pass the course.

As per university policy, if you do not sit the final exam, you automatically fail the course. Please check the dates now to make sure that you have no unavoidable conflicts.

Here is the schedule:
  • Midterm 1: Monday February 14 (in class for both sections)
  • Midterm 2: Friday April 1 (in class for both section)
  • Final exam: Tuesday May 10 (LEC 001: 3pm-6pm, LEC002: 7pm-10pm)

Calculators and notes will NOT be allowed for the exams.

To obtain full credit for an exam question, you must obtain the correct answer and give a correct and readable derivation or justification of the answer. Unjustified correct answers will be regarded very suspiciously and will receive little or no credit. The graders are looking for demonstration that you understand the material. To maximize credit, cross out incorrect work. We will be scanning all exams so you will get them back electronically.

After each midterm, there will be a brief window when you can request a regrade. If you are unsure about making a regrade request consult myself or your GSI beforehand. Regrade requests may result in a lowering of your grade. As per university policy, final exams cannot be regraded.

DSP students requiring accommodations for exams will be contacted ahead of time with alternative arrangements.

Cheating is unacceptable. Any student caught cheating will be reported to higher authorities for disciplinary action.

Here is a link to a collection of practice exams. Solutions will be posted as we progress through the course.

As the semester progresses the tabs on the table below will open, linking to detailed pre-recorded videos and lecture notes. These will essentially be identical to the in-person lectures.

I taught this course in Spring 2018 and here is a link to the videos (and notes) I recorded then. Based on the feedback I got then, we will do things a little differently this time, with more of an emphasis on examples and geometric intuition. These videos are definitely worth looking at, but do be aware they will not always match with the videos below and you may miss certain things if you rely on them alone.

WhenWhat Where
Week 1 (1/18 - 1/21) 1.1, 1.2
Week 2 (1/24 - 1/28) 1.3
1.4, 1.5
1.5, 1.7
Week 3 (1/31 - 2/4) 1.8, 1.9
Week 4 (2/7 - 2/11) 2.2, 2.3
3.1, 3.2
Week 5 (2/14 - 2/18)
4.1, 4.2
4.1, 4.3, 4.5
Week 6 (2/22 - 2/25) 4.3, 4.5, 4.6
4.4, 4.7
Week 7 (2/28 - 3/4) 5.1, 5.2
Week 8 (3/7 - 3/11) 5.4
Week 9 (3/14 - 3/18) 6.2
Week 10 (3/21 - 3/25)
Week 11 (3/28 - 4/1) 6.5
Week 12 (4/4 - 4/8) 7.1
Week 13 (4/11 - 4/15) 4.2, 4.3
4.4, 4.5
Week 14 (4/18 - 4/22) 9.1, 9.4
9.5, 9.6
Week 15 (4/25 - 4/29) 10.3, 10.4
10.3, 10.4
Week 16 (5/2 - 5/6)
Week 17 (5/9 - 5/13)

Grades are calculated as follows:

Homework 10%
Quizzes 10%
First Midterm 20%
Second Midterm 20%
Final Exam 40%

If your final exam score is higher than your lowest midterm exam score, then it will replace it, thus accounting for 60% of your grade. This means you can miss a midterm for any reason whatsoever and it will not necessarily adversely effect your grade.

Your final letter grade will ultimately be decided by your ability to demonstrate a crisp understanding of the material and the ability to apply it to a diverse set of problems. Broadly speaking I will be looking for the following criteria for each letter grade:

  • A-/A/A+: A clear demonstration that the central concepts have been fully understood; Computational techniques (and their many subtleties) have been mastered and can be applied accurately to a diverse problem set; A strong understanding of how the abstract concepts can be applied to many real world applications.
  • B-/B/B+: Demonstration that the central concepts have been reasonably understood, but perhaps with minor misunderstandings; Core computational techniques have been reaonably understood (but generally not key subtleties) and can be applied fairly accurately to a fairly large problem set; Reasonable understanding of how the abstract concepts can be applied to some real world applications.
  • C-/C/C+: Demonstration that the central concepts have been vaguely understood, but with major misunderstandings; Core computational techniques have been poorly understood and can be a applied accurately only in the most standard examples; Weak understanding of how the abstract concepts can be applied to even basic real world applications.

To be as fair as possible, I will also take into account the historic average of the class. This means that if I set an exam which is very difficult it will be taken into account in the final letter grades.

Please note: incomplete grades, according to university policy, can be given only if unanticipated events beyond your control (e.g. a medical emergency) make it impossible for you to complete the course, and if you are otherwise passing (with a C- or above).

Enrollment: For question about enrollment contact Marsha Snow, or email

The Student Learning Center provides support for this class, including full adjunct courses, review sessions for exams, and drop-in tutoring. This is a truly fantastic resource. I definitely recommend you take advantage of it.