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Alexander Paulin |
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me | research | teaching | CV
Linear Algebra Math 56 Fall 2023
Welcome to Math 56! This is a brand 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. 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!
9.30am-11am TT, Stanley Hall 105
Attendance is not mandatory (although, I'd strongly advise that you go). In-person lectures will recorded and be posted below.
Here are the in-person lecture notes and handouts.
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'm going to make the text available in different parts as we progress though the semester. Any feedback would be much appreciated. There will undoubtedly be times that I'll post an updated version of a given chapter or part. I'll let you know when this has happened and what changes have been made. It's most likely to be the addition of more 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.
Chapters 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, and 14 (Updated 11/26)
Every MWF you will have a one hour discussion session led by a GSI (graduate student instructor). Your first discussion section will be on Friday August 25. 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. These problems will be carefully chosen from the textbook. 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 Mathemtatics 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 following link. Attendance at discussion section is not mandatory, although I'd strongly advise you to go. The problems you'll be going through in these sessions will be specifically chosen by me to deepen your understanding. Not being there is a bad plan.
You can only attend the section you are officially enrolled in. If you have enrollment questions contact: enrollment@math.berkeley.edu
Here is contact information for all GSIs.
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 12pm-1pm Monday,1pm-2pm Wednesday,and 2pm-3pm Friday 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.
Here is the homework schedule.
Homework assignments are due each Tuesday at 11.59pm on Gradescope. Here are instructions for how to upload your work. If you have issues submitting your work, contact your GSI. Your lowest two homework scores will be dropped.
Assignments will be graded on completeness. 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 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 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 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.
Homework Solutions will be made available shortly after submission at this link.
In addition to homework, roughly every two to three weeks you will have a project.
Projects will give you the opportunity to explore certain topics in more depth. Think about them like an extended homework problem. They will be due on Friday, roughly every two weeks on Gradescope. Here are instructions on how to upload your work. If you have issues submitting your work, contact your GSI. Note that unlike the homework, you'll need to submit a pdf of fixed length following the project template. Your lowest score will be dropped from your grade.
The projects will gradually be made available at this link.
| What | When |
|---|---|
| Project 1 | 9/8 |
| Project 2 | 9/22 |
| Project 3 | 10/6 |
| Project 4 | 10/27 |
| Project 5 | 11/10 |
| Project 6 | 12/1 |
Here a link that will contain practice exams. These will be very similar to the actual exams. Once they are posted, you should study them very carefully
There will be one midterm and a final. There will be no make-up exams, unless there are truly exceptional circumstances.
As outlined in the course policy below, if your final exam score is higher than the midterm score it will replace it. That measn you could miss the midterm altogether and still be fine. I certainly wouldn't recommend it though.
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: Tuesday October 10th, in class (one hour exam)
- Final Exam: Tuesday December 12th, 3pm-6pm (thee hour exam) Location TBD
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 the exams, 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. Frivolous regrade requests may result in a lowering of your grade.
DSP students requiring accommodations for exams must submit to the instructor a "letter of accommodation" from the Disabled Students Program. Due to delays in processing, you are encouraged to contact the DSP office before the start of the semester. You will be contacted shortly before exams with specific exam logistics related to your accomodations.
Cheating is unacceptable. Any student caught cheating will be reported to higher authorities for disciplinary action. Don't do it. It not remotely worth it.
Below is the week by week schedule for the course. As the course progress the tabs will become active and you'll be able to access videos by clicking on them.
| When | What | Where |
|---|---|---|
| Week 1 (8/21 - 8/25) | 1.1 | |
| Week 2 (8/28 - 9/1) | 1.2 | |
| 1.3 | ||
| Week 3 (9/4 - 9/8) | 2.1, 2.2 | |
| 2.3 | ||
| Week 4 (9/11 - 9/15) | 3.1 | |
| 3.2 | ||
| Week 5 (9/18- 9/22) | 3.3 | |
| 4.1, 4.2 | ||
| Week 6 (9/25 - 9/29) | 4.3 | |
| 5.1 | ||
| Week 7 (10/2 - 10/6) | 5.2 | |
| 6.1, 6.2 | ||
| Week 8 (10/9 - 10/13) |
| |
| 7.1 | ||
| Week 9 (10/16 - 10/20) | 7.2, 7.3 | |
| 8.1, 8.2 | ||
| Week 10 (10/23 - 10/27) | 9.1, 9.2 | |
| 10.1, 10.2 | ||
| Week 11 (10/30 - 11/3) | 11.1 | |
| 11.2 | ||
| Week 12 (11/6- 11/10) | 12.1 | |
| 12.2 | ||
| Week 13 (11/13 - 11/17) | 13.1 | |
| 13.2 | ||
| Week 14 (11/20 - 11/24) |
| 14.1 |
| Week 15 (11/27 - 12/1) | 14.1 | |
| 14.2 | ||
| Week 16 (12/4 - 12/8) | ||
| Week 17 (12/11 - 12/15) |
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Grades are calculated as follows:
| Homework | 15% |
| Projects | 15% |
| Midterm | 30% |
| Final Exam | 40% |
If your final exam score is higher than your midterm exam score, then it will replace it, thus accounting for 70% of your grade. This means you can miss the 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.