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Alexander Paulin |
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me | research | teaching | CV
Linear Algebra Math 56 Fall 2025
Welcome to Math 56!
This is a relatively new course here at Berkeley and I'm excited for you to be part of it. Math 56 is a first course in linear algebra, one of the most fundamental subjects in mathematics. If this is your first experience with university level mathematics outside of calculus, this class will open your eyes to a whole new way of thinking. Concepts with vague intutive meaning, like dimension, will become precise and powerful. The techniques that you will learn in this class are some of the most widely applied in all of the sciences. We will tilt towards applications in data scicence, showing why linear algebra is at the forefront of modern computing and AI. By the end of the course you will know why!
Lectures are where I will introduce and explore the material. Throughout the semester, I'll do my best to make you part of the process, giving you the chance to try problems yourself and ask questions. If there's something you feel unsure of, don't feel nervous about putting your hand up. If you are thinking it, then so are lots of other people in the room. I'm here to guide you towards proper understanding, not deliver dry uninterupted monologues. The more like a back and forth conversation, the better!
12.30pm-2.00pm TuTh, 2050 Valley Life Sciences Building
As is tradition at Berkeley, the lecture will formally begin ten minutes after the official start time. That gives you enough time to get across campus if you need to. Attendance at lecture is not mandatory, although I strongly advise that you go.
All lectures will come with pre-recorded videos and lecture notes. You can find these in the Schedule and Course Materials tab below. In terms of content, these will be identical to in-person lectures.
I strongly recommend against using the pre-videos as a total replacement for in-person lectures
In-person lectures offer a more focused learning experience, allowing you to ask and hear questions in real time. While videos are of course great for reviewing material at your own pace, it's challenging to stay fully engaged during an initial viewing. The temptation to skip through sections will lead to missing important details, something that's less likely to occur when we are all together in person.
Here's some advice about getting the most out of lectures and the pre-recorded videos.
- Pay attention to the details. These lectures will likely be in more depth than what you are used to. Simply grasping the basic idea of a concept won't suffice. Your goal should be to comprehend each concept in granular detail. Every point covered will be crucial to the topic, so if something isn't clear, don't ignore it. Ask questions in class, or jot down your questions to discuss them during office hours or in the discussion section later on. Understanding these nuances is key to mastering the material.
- Take your own notes. I'll be writing everything out in real-time during lecture. If you are following along with me doing the same, it'll really help you focus. A good tip is to highlight points that weren't clear to you so you can specifically revist them later.
- Come back to the videos after the in-person lecture. Video lectures offer great flexibility for learning. Being able to revisit them allows you to reinforce your understanding of complex topics or clarify any confusion you may have. This ability to review at your own pace can significantly enhance comprehension and retention of the material. Make use of it, but don't let it completely replace in-person lectures.
- Be very cautious of other math resources online. This course is structured to be as self-contained as possible. If you find yourself struggling with a concept, revisit the corresponding video lecture rather than turning to random online sources. Such sources almost always lack sufficient detail and can lead to a false sense of understanding. Additionally, avoid relying on tools like ChatGPT. They won't contribute to your learning, especially since AI is still terrible at mathematics.
- Make sure you understand the examples. The exercises in the worksheets and homework are closely related to examples I go through in the lecture. Realy make sure you really understand the approach I'm taking to solving problems in the lectures.
- Don't skip any lectures. Even if the topic is familiar to you, there will be subtleties that you have likely not seen before. Think of it like reading a story. If you skip a paragraph you'll get lost very quickly.
You can find links to the lecture videos and notes in the Schedule and Course Materials tab below.
We are using a new textbook for this course. In fact it's so new it's not been published yet! That's because I'm the one writing it along with my two coauthors, David Nadler and Vineet Gupta. We've been working on it for almost two years and it's now in a polished, near complete state. Unlike most other classes, this textbook was written specifically for Math 56. It's structured to sync with the weekly lecture format, and will lead you through the subject in a intuitive visual way. Throughout the text are example questions with accompanying solutions, so you can really see the nuts and bolts of the subject. These all come with a prompt for you to try a similar question. These are what you'll be looking at in discussion section. Homework questions will be taken from the end of section exercises.
I strongly recommend that you read the textbook throughout the semester. It's going to sync very closely with the lectures and will explain details we won't necessarily be able to cover in class.
Every week you will have three hours of discussion session led by a GSI (graduate student instructor). These sessions provide an excellent opportunity to explore the material as a group, with each section consisting of around 40 students. In these interactive discussions, you will collaborate with your peers to tackle challenging problems, enabling you to truly grasp and excel in the subject matter. Here are some tips to get the most out of these sessions.
- Mastering Mathematics is a challenge and most challenges are best approached as a group. Regard your discussion section not only as an opportunity to understand the course material better but also as a place to connect with classmates for collaborative work outside of class. Having other people to bounce ideas off is incredibly valuable. The more perspectives on a difficult problem the better.
- Don't feel apprehensive about participating. Many of of us get nervous speaking in public, doubly so when it comes to mathematics. You may think that only you are struggling with a concept, but that is never ever true. If something's confusing, ask questions. If the instructor asks a question, try answering it. It can be scary at first but over time it won't be and the payoff will be enormous in the long run
- Be active in group work. Your will frequently be split 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.
- Be there. This is probably the most important of all. Even if you feel that you aren't getting much out discussion section, you are. It's three whole hours a weeks where you are completely focused on the class. Ask yourself the following question: Am I realy going to spend three hours a week going through the worksheets on my own? For me the answer would be no! That's a lot of time to lose and it adds up over the course of a semester. Not attending section is almost always a direct indicator of heading for a bad grade.
You can only attend the section you are officially enrolled in. If you have enrollment questions contact: enrollment@math.berkeley.edu
Attendance at discussion section is not mandatory, although I strongly advise that you go as I've explained above.
Here is contact information for all GSIs.
You'll be following fixed worksheets in your discussions. They'll be posted in the Schedule and Course Materials tab below.
In general, if you have any queries or issues relating to the course, the first person to contact directly is your GSI.
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.
I'll be having office hours every 3.00pm-4.30pm every Tuesday and Thurday in 796 Evans Hall. As a default GSI office hours are in-person, although some may be conducted online via Zoom.
Here is a link with the full office hour schedule.
There will be weekly homework assignments, each covering topics from the previous week. These will be due each Wedesday at 11.59pm on Gradescope (an online platform to submit assignments). You can access Gradescope from this link. Here are instructions for how to upload your work. If you have issues submitting your work, contact your GSI.
The homework exercises will be from the textbook. Here is a link to the homework schedule. You can also link to this directly from the Schedule and Course Materials tab below.
Homework Solutions will be made available shortly after submission in the Schedule and Course Materials Tab.
Each assignment will be graded on a combination of completeness and correctness. More specifically you will get three points for submitting solutions to all exercises. In addition, each week a randomly selected question will be graded (out of three points) for correctness. 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 and office hours 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 much 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 mathematics classes 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 may struggle to complete the homework assignment if you start on the night before it is due. Work consistently in small installments.
- 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. Make sure you attempt them. Almost every problem will be very similar to one done in the video lectures or the workseets. Consult those if you're unsure of where to start.
- Seek help, but only after seriously thinking about a problem on your own. If you're struggling with a challenging problem you should spend at least 30 minutes on your own thinking about it carefully. Even if you fail to make a breakthrough, this is still more worthwhile than giving up after a couple of minutes and talking to peer or looking up a random source on the internet. Knowing what doesn't work is just as important knowing what does. The questions will be very closely related to the lectures and textbook. Those should be where you look for help first. In future courses you'll be solving problems that require days (or weeks) of dedicated thought. Now is the time to hone this skill and get comfortable with not being able to immediately solve a problem.
In addition to homework, roughly every week, you will have either a quiz or project.
Quizzes will take place on Friday in discussion section. They will last about 15 minutes and be closely related to the homework submitted that week. Your lowest score will be dropped from your grade.
Projects will give you the opportunity to explore certain topics in more depth, perhaps exploring interesting real-world applications, or more subtle questions what we don't have time to cover in lectures. They will be due on Friday evenings. Here are instructions on how to upload your work. Your lowest score will be dropped from your grade.
The solutions to the projects will be available after submission in the Schedule and Course Materials tab below.
| What | When |
|---|---|
| Quiz | Week 2 |
| Project - Sparse Vectors | Week 3 |
| Project - Linear Classifiers | Week 4 |
| Quiz | Week 5 |
| Project - Computer Graphics | Week 6 |
| Quiz | Week 7 |
| Project - Neural Networks | Week 9 |
| Quiz | Week 10 |
| Project - Markov Chains | Week 11 |
| Project - Areas in Higher Dimensions | Week 13 |
| Quiz | Week 15 |
There will be no make-up quizzes, unless there are exceptional circumstances.
There will be two (1-hour) midterm exams and one (3-hour) final exam.
Following the course policy (see below), 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 exam schedule.
- Midterm 1: Tuesday October 14 (in class)
- Midterm 2: Thursday November 13 (in class)
- Final Exam: Friday December 19 • 8:00AM - 11:00AM (Location TBD)
Calculators are NOT be allowed for the exams. That's the policy for all university math classes. That may seem scary, but it means the exams are designed not to require challenging mental arithmetic. Thats a good thing!
Cheatsheets are not be allowed for the exams. Making one yourself is a very good way of preparing, but it can't be taken into the test.
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.
All special exam accomodations for DSP students will be handled by the DSP proctoring service. If you are a DSP student with exam accomodations you should contact them at least a week ahead of the exam to make arrangments.
It goes without saying that cheating is unacceptable. Any student caught cheating will be reported to higher authorities for disciplinary action.
There will be no make-up exams, unless there are truly exceptional circumstances.
Here is my basic advice on how to do well in exams.
- Take a moment to look through the whole exam at the start. We can all agree that exams are no fun. I have always been very anxious taking exams and something that always helped was looking through the exam for a few minutes at the start. Find a question you know how to approach and tackle that first. It'll help settle your nerves.
- Success is in the details. It's not enough that you have a broad superficial understanding of a topic. You must appreciate the subtleties and how they turn up in problems.
- Diagnose and solve systematically. Once you've identified what type of problem you are dealing with, think about what's in your toolkit. How does one approach such a problem in a methodical way? You want to be familiar and comfortable with the standard procedures to deal with different problems. This is a skill for life and is the essence of analytic problem solving.
- The practice exams are the single most important component of exam preparation. I'm going to provide you with three practice exams for each test. These will be pretty challenging. You should be aiming for mastery of every question in the practice exams before the real thing. The main exam will be close variations of these questions and if you've not spent serious time with them it is going to be much harder. Ideally, by the time you take the main exam you want to be able to write perfect solutions to the practice questions unprompted. You should be looking at these at least one week before the main exam. Solutions to practice exams will be provided one week before the main exams.
Lastly, I know that many of you find exams acutely stressful (I know I do). If you are struggling (before or during the exam) do come and speak to me or a GSI. We are here to help.
This is your one-stop-shop for all the course materials: videos, notes, worksheets, etc. As the semester progresses each of the links below will go live. For example, video lectures will be available directly after the LEC002 in-person lecture is complete.
| When | What | Where | |
|---|---|---|---|
| Week 1 (8/25 - 8/29) | 1.1 | ||
| Week 2 (9/1 - 9/5) | 1.2, 1.3 | ||
| 1.4 | |||
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| Week 3 (9/8 - 9/12) | 1.5 | ||
| 2.1 | |||
| Week 4 (9/15 - 9/19) | 2.2 | ||
| 2.3 | |||
| Week 5 (9/22- 9/26) | 3.1 | ||
| 3.2 | |||
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| Week 6 (9/29 - 10/3) | 4.1 | ||
| 4.2 | |||
| Week 7 (10/6 - 10/10) | 5.1, 5.2 | ||
| 6.1 | |||
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| Week 8 (10/13 - 10/17) |
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| 7.1, 7.2 | |||
| Week 9 (10/20 - 10/24) | 8.1, 8.2 | ||
| 9.1, 9.2 | |||
| Week 10 (10/27 - 10/31) | 10.1 | ||
| 10.2 | |||
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| Week 11 (11/3 - 11/7) | 11.1, 11.2, 11.3 | ||
| 12.1 | |||
| Week 12 (11/10 - 11/14) |
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| Week 13 (11/17 - 11/21) | 12.2 | ||
| 13.1 | |||
| Week 14 (11/24 - 11/28) |
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| Week 15 (12/1 - 12/5) | 13.2 | ||
| 13.2 | |||
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| Week 16 (12/8 - 12/12) | |||
| Week 17 (12/15 - 12/19) |
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Grades are calculated as follows:
| Homework | 10% |
| Quizzes | 5% |
| Projects | 5% |
| Midterm 1 | 20% |
| Midterm 2 | 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 univlersity 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 enrollment@math.berkeley.edu.