Content: We will closely follow the ouline of Gil Strang's linear algebra course taught at MIT. Your introduction to the material will come from readings as well as video lectures; the details will depend on the week.  I will hold and introductory lecture as well as office hours to discuss the material;  you should be ready with questions.  Each week's work will be detailed as one of the modules appearing below.

Book and Materials: For a good portion of the class, we will follow the outline of MIT's linear algebra course from: [link]. You will find all the course videos therein. The following website contains a variety of resources for the class, including answers to homework questions as well as copies of old exams: [link]. The formal book for the class is Introduction to Linear Algebra by Gilbert Strang. I will use the fourth edition, which is readily available used as well as a [PDF]. Make sure to secure your own copy.

Reading and watching mathematics: Learning mathematics is not a spectator sport. Reading mathematics is not like reading a novel; watching mathematics is not like watching an action thriller. Some paragraphs are easy to digest, but you may find yourself looking at one line of text for five or more minutes trying to understand what the author is trying to say.  Use the pause button when watching a video. As you read or watch, take notes, just as you do in class. This is crucial! If questions arise, write them down and ask during office hours. In each module, I will let you know how long you should expect to spend reading and watching the material.  Some weeks, this will be a substantial commitment of time even before you start the homework.

Office hours: I will hold an office hour each week via Zoom.  The scheduled times as well as links to join will appear in each week's module.  I am also happy to talk with you at a different time; just let me know and we can schedule a meeting.

Homework: Your homework will be graded using [GradeScope]; To begin, you will need to set up an account.  I will send you the code for our class by email.  Each homework assignment will need to be submitted as a .pdf file.  If you typeset your homework using LateX, this is straightforward.  If you write-up your homework the old-fashioned way using pencil and paper, use a scanner or a phone scanner app.  See the GradeScope help document for a list of suggestions [pdf]. Please let me know if this does not work for you; I will come up with an alternative.

Collaboration and Groups: Throughout the course of the class, you will be a part of a group. While your work will be written-up individually, I would like all of you to check in and discuss the course material and homework twice a week. The meetings will vary in length, but each time you meet, I would like a short, one-paragraph report from one member of each group.

Grades: Your grade will be based on the homework (25%), exams and final (60%), and class participation (15%). I will drop your lowest three homework scores. The exams and final will be flexible take-home affairs.

Feedback: I expect the nature of instruction in the class to be an evolving paradigm.  Let me know what things work for you and which ones do not. The sooner, the better.

In my lecture this week, I will give an overview of some of the modern applications of linear algebra, the slides for your reference are here: [PDF]. We will meet by Zoom at 1pm on Monday and I will hold an office hour on Thursday at 4pm.  Here are the links:

[Monday 1pm]

[Thursday 4pm]

If you would like to meet at any other point, please send me an email. The course itself will begin more humbly with a review of vectors and an introduction to systems of linear equations. Take good notes: even the material will not seem completely new, I like Strang's perspective and it will serve you well to know it well.

Assignment: There are two parts this week.  The first covers Sections 1.1, 1.2, and 2.1 and is to be completed by Thursday at noon. The second covers Section 1.3 and is to be completed by Monday at noon.

• Read sections 1.1, 1.2, and 2.1 or watch the video: [link]. Take good notes, you will submit them as the first part of your homework.
• Complete the Check Yourself questions and compare with the provided solutions.
• The homework problems are as follows. 
  
• Submit a copy of your notes for these sections.
• §1.1 22, 31*
• §1.2 8, 27, 28*, 30*
• §2.1 4, 6, 9, 10, 15, (16), (17), 26, 29, 31*
A note on my notation. The starred problems are challenges. Do as many as you have time and energy for. The parenthetical problems are included because I think they are important, but for humanitarian reasons, I am not making them part of the official homework. Work on them for your own edification, but don't turn them in.

• After working on the problems by yourself, check in with your group. Discuss the material and the problems. Afterwards, designate someone to send me an update on your progress.  What was the most challenging part of the assignment? What parts of the lectures are still unclear?
• You should submit your homework solutions as a single .pdf file; a nice way to do create this document is to use a [phone scanner app]. This part of the homework is due on Thursday at 12pm. Submit it via [GradeScope].


Part 2


• Read section 1.3 or watch the video: [link]. Take good notes, you will submit them as the first part of your homework.
• The homework problems are as follows:
• Submit a copy of your notes for this section.
• §1.3 3, 4, 6, 7, 14*
• After working on the problems by yourself, check in with your group. Discuss the material and the problems. Afterwards, designate someone to send me an update on your progress.  What was the most challenging part of the assignment? What parts of the lectures are still unclear?
• You should submit your solutions as a single .pdf file; a nice way to do create this document is to use a [phone scanner app]. Homework is due on Monday at 12pm. Submit it via [GradeScope].:

This week's homework contained a short problem about the arithmetic-geometic mean inequality. It is a very useful result, but we may not see it again in the course. The following clip gives one good reason for using geometric instead of arithmetic means, and outlines my favorite proof of this result. It ends with a few exercises for the bold.

AMGM Inequality

[PDF]

In my lecture this week, I summarized the main goals of each of the sections of this week's reading. The slides for your reference are here: [PDF]. We will meet by Zoom at 1pm on Monday and I will hold an office hour on Thursday at 4pm.  Here are the links:

[Monday 1pm]

[Thursday 4pm]

If you would like to meet at any other point, please send me an email.

Assignment: There are two parts this week.  The first covers Sections 2.2 and 2.3 and is to be completed by Thursday at noon. The second covers Sections 2.4 and 2.5 and is to be completed by Monday at noon.

• Read sections 2.2 and 2.3 or watch the video: [link]. Take good notes, you will submit them as the first part of your homework.
• Complete the Check Yourself questions and compare with the provided solutions.
• The homework problems are as follows. 
  
• Submit a copy of your notes for these sections.
• §2.2 5, 11, 12, 13, 20, 32*
• §2.3 1, 7, 8, 18, 25, 28
• After working on the problems by yourself, check in with your group. Discuss the material and the problems. Afterwards, designate someone to send me an update on your progress.  What was the most challenging part of the assignment? What parts of the lectures are still unclear?
• You should submit your homework solutions as a single .pdf file; a nice way to do create this document is to use a [phone scanner app]. This part of the homework is due on Thursday at 12pm. Submit it via [GradeScope].


Part 2


• Read sections 2.4 and 2.5 or watch the video: [link]. Take good notes, you will submit them as the first part of your homework.
• The homework problems are as follows:
• Submit a copy of your notes for these sections.
• §2.4 5, 6, 13, 14, 23, 32, 36*
• §2.5 1, (6), 7, 10, (11), 15, 22, 24
• After working on the problems by yourself, check in with your group. Discuss the material and the problems. Afterwards, designate someone to send me an update on your progress.  What was the most challenging part of the assignment? What parts of the lectures are still unclear?
• You should submit your solutions as a single .pdf file; a nice way to do create this document is to use a [phone scanner app]. Homework is due on Monday at 12pm. Submit it via [GradeScope].:

In my lecture this week, I summarized the main goals of each of the sections of this week's reading. The slides for your reference are here: [PDF]. We will meet by Zoom at 1pm on Monday and I will hold an office hour on Thursday at 4pm.  Here are the links:

[Monday 1pm]

[Thursday 4pm]

If you would like to meet at any other point, please send me an email.

Assignment: There are two parts this week.  The first covers Sections 2.6 and is to be completed by Thursday at noon. The second covers Section 2.7 and is to be completed by Monday at noon.

• Read section 2.6 or watch the video: [link]. Take good notes, you will submit them as the first part of your homework.
• The homework problems are as follows. 
  
• Submit a copy of your notes for this section.
• §2.6 7, 9, 11, 13, 17, 18, 23
• After working on the problems by yourself, check in with your group. Discuss the material and the problems. Afterwards, designate someone to send me an update on your progress.  What was the most challenging part of the assignment? What parts of the lectures are still unclear?
• You should submit your homework solutions as a single .pdf file; a nice way to do create this document is to use a [phone scanner app]. This part of the homework is due on Thursday at 12pm. Submit it via [GradeScope].


Part 2


• Read section 2.7 or watch the video: [link]. Take good notes, you will submit them as the first part of your homework.
• The homework problems are as follows:
• Submit a copy of your notes for this section.
• §2.7 4, 9, 12, (13), 16, 20, 22, 26, (29), (36), 39*, 40*
• After working on the problems by yourself, check in with your group. Discuss the material and the problems. Afterwards, designate someone to send me an update on your progress.  What was the most challenging part of the assignment? What parts of the lectures are still unclear?
• You should submit your solutions as a single .pdf file; a nice way to do create this document is to use a [phone scanner app]. Homework is due on Monday at 12pm. Submit it via [GradeScope].:

In my lecture this week, I summarized the main goals of each of the sections of this week's reading. The slides for your reference are here: [PDF]. We will meet by Zoom at 1pm on Monday and I will hold an office hour on Thursday at 4pm.  Here are the links:

[Monday 1pm]

[Thursday 4pm]

If you would like to meet at any other point, please send me an email.

Assignment: There are two parts this week.  The first covers Sections 3.1 and is to be completed by Thursday at noon. The second covers Section 3.2 and is to be completed by Monday at noon.

• Read section 3.1 or watch the video: [link]. Take good notes, you will submit them as the first part of your homework.
• The homework problems are as follows. 
  
• Submit a copy of your notes for this section.
• §3.1 4, 9, 14a, 15c, 18a, 20, 23, 25, 26, 30*
• After working on the problems by yourself, check in with your group. Discuss the material and the problems. Afterwards, designate someone to send me an update on your progress.  What was the most challenging part of the assignment? What parts of the lectures are still unclear?
• You should submit your homework solutions as a single .pdf file; a nice way to do create this document is to use a [phone scanner app]. This part of the homework is due on Thursday at 12pm. Submit it via [GradeScope].


Part 2


• Read section 3.2 or watch the video: [link]. Take good notes, you will submit them as the first part of your homework.
• The homework problems are as follows:
• Submit a copy of your notes for this section.
• §3.2 1-3, 9, 18, 19, 37*
• After working on the problems by yourself, check in with your group. Discuss the material and the problems. Afterwards, designate someone to send me an update on your progress.  What was the most challenging part of the assignment? What parts of the lectures are still unclear?
• You should submit your solutions as a single .pdf file; a nice way to do create this document is to use a [phone scanner app]. Homework is due on Monday at 12pm. Submit it via [GradeScope].:

In my lecture this week, I summarized the main goals of each of the sections of this week's reading. The slides for your reference are here: [PDF]. We will meet by Zoom at 1pm on Monday and I will hold an office hour on Thursday at 4pm.  Here are the links:

[Monday 1pm]

[Thursday 4pm]

If you would like to meet at any other point, please send me an email.

Assignment: There are two parts this week.  The first covers Sections 3.3 and 3.4 and is to be completed by Thursday at noon. The second covers Section 3.5 and is to be completed by Monday at noon.

• Read sectios 3.3 and 3.4 or watch the video: [link]. Take good notes, you will submit them as the first part of your homework.
• The homework problems are as follows. 
  
• Submit a copy of your notes for these sections.
• §3.3 1, 7, 10, 12, 15, 19, 20, 23
• §3.4 1, 3, 8, 13, 21, 36*
• After working on the problems by yourself, check in with your group. Discuss the material and the problems. Afterwards, designate someone to send me an update on your progress.  What was the most challenging part of the assignment? What parts of the lectures are still unclear?
• You should submit your homework solutions as a single .pdf file; a nice way to do create this document is to use a [phone scanner app]. This part of the homework is due on Thursday at 12pm. Submit it via [GradeScope].


Part 2


• Read section 3.5 or watch the video: [link]. Take good notes, you will submit them as the first part of your homework.
• The homework problems are as follows:
• Submit a copy of your notes for this section.
• §3.5 2, 5, 11, 14, 16, 21, 23, 26, 43*, 45*
• After working on the problems by yourself, check in with your group. Discuss the material and the problems. Afterwards, designate someone to send me an update on your progress.  What was the most challenging part of the assignment? What parts of the lectures are still unclear?
• You should submit your solutions as a single .pdf file; a nice way to do create this document is to use a [phone scanner app]. Homework is due on Monday at 12pm. Submit it via [GradeScope].:

In my lecture this week, I summarized the main goals of this week's reading: [PDF]. We will meet by Zoom at 1pm on Monday and I will hold an office hour on Thursday at 4pm.  Here are the links:

[Monday 1pm]

[Thursday 4pm]

If you would like to meet at any other point, please send me an email.

Assignment: There is just one part this week, to be completed by Thursday at Noon. The second part will consist of an exam, and some optional reding, to be completed by Monday at Noon.

• Read section 3.6 or watch the video: [link]. Take good notes, you will submit them as the first part of your homework.
• The homework problems are as follows. 
  
• Submit a copy of your notes for this section.
• §3.6 2, 11, 13, 24, 25, 26, 30*
• After working on the problems by yourself, check in with your group. Discuss the material and the problems. Afterwards, designate someone to send me an update on your progress.  What was the most challenging part of the assignment? What parts of the lectures are still unclear?
• You should submit your homework solutions as a single .pdf file; a nice way to do create this document is to use a [phone scanner app]. This part of the homework is due on Thursday at 12pm. Submit it via [GradeScope].


Part 2


• Optional: Read section 8.2 or watch the video: [link].
• On Thursday at 1pm, or perhaps a few minutes before, I will send you a copy of the first exam. It will be a take-home, and you will have 24-hours to complete it. Detailed directions will come with the exam. You should submit your solutions as a single .pdf file; a nice way to do create this document is to use a [phone scanner app]. The exam is due on Monday at 12pm. Submit it via [GradeScope].:

In my lecture this week, I summarized the main goals of each of the sections of this week's reading. The slides for your reference are here: [PDF]. We will meet by Zoom at 1pm on Monday and I will hold an office hour on Thursday at 4pm.  Here are the links:

[Monday 1pm]

[Thursday 4pm]

If you would like to meet at any other point, please send me an email.

Assignment: There are two parts this week.  The first covers Section 4.1 and is to be completed by Thursday at noon. The second covers Section 4.2 and is to be completed by Monday at noon.

• Read section 4.1 or watch the video: [link]. Take good notes, you will submit them as the first part of your homework.
• The homework problems are as follows. 
  
• Submit a copy of your notes for this section.
• §4.1 9, 11, 15, 21, 23, 24, 25, 28
• After working on the problems by yourself, check in with your group. Discuss the material and the problems. Afterwards, designate someone to send me an update on your progress.  What was the most challenging part of the assignment? What parts of the lectures are still unclear?
• You should submit your homework solutions as a single .pdf file; a nice way to do create this document is to use a [phone scanner app]. This part of the homework is due on Thursday at 12pm. Submit it via [GradeScope].


Part 2


• Read section 4.2 or watch the video: [link]. Take good notes, you will submit them as the first part of your homework.
• The homework problems are as follows:
• Submit a copy of your notes for this section.
• §4.2 1, 5, 11, 13, 16, 17, 21, 22, 33*
• After working on the problems by yourself, check in with your group. Discuss the material and the problems. Afterwards, designate someone to send me an update on your progress.  What was the most challenging part of the assignment? What parts of the lectures are still unclear?
• You should submit your solutions as a single .pdf file; a nice way to do create this document is to use a [phone scanner app]. Homework is due on Monday at 12pm. Submit it via [GradeScope].:

In my lecture this week, I summarized the main goals of each of the sections of this week's reading. The slides for your reference are here: [PDF]. We will meet by Zoom at 1pm on Monday and I will hold an office hour on Thursday at 4pm.  Here are the links:

[Monday 1pm]

[Thursday 4pm]

If you would like to meet at any other point, please send me an email.

Assignment: There are two parts this week.  The first covers Section 4.3 and is to be completed by Thursday at noon. The second covers Section 4.4 and is to be completed by Monday at noon.

• Read section 4.3 or watch the video: [link]. Take good notes, you will submit them as the first part of your homework.
• The homework problems are as follows. 
  
• Submit a copy of your notes for this section.
• §4.3 17-23, 25*,28*,29*
• After working on the problems by yourself, check in with your group. Discuss the material and the problems. Afterwards, designate someone to send me an update on your progress.  What was the most challenging part of the assignment? What parts of the lectures are still unclear?
• You should submit your homework solutions as a single .pdf file; a nice way to do create this document is to use a [phone scanner app]. This part of the homework is due on Thursday at 12pm. Submit it via [GradeScope].


Part 2


• Read section 4.4 or watch the video: [link]. Take good notes, you will submit them as the first part of your homework.
• The homework problems are as follows:
• Submit a copy of your notes for this section.
• §4.4 4, 6, 10, 11, 20, 22, 23, 34
• After working on the problems by yourself, check in with your group. Discuss the material and the problems. Afterwards, designate someone to send me an update on your progress.  What was the most challenging part of the assignment? What parts of the lectures are still unclear?
• You should submit your solutions as a single .pdf file; a nice way to do create this document is to use a [phone scanner app]. Homework is due on Monday at 12pm. Submit it via [GradeScope].:

In my lecture this week, I summarized the main goals of each of the sections of this week's reading. The slides for your reference are here: [PDF]. We will meet by Zoom at 1pm on Monday and I will hold an office hour on Thursday at 4pm.  Here are the links:

[Monday 1pm]

[Thursday 4pm]

If you would like to meet at any other point, please send me an email.

Assignment: There are two parts this week.  The first covers Section 5.1 and is to be completed by Thursday at noon. The second covers Section 5.2 and is to be completed by Monday at noon.

• Read section 5.1 or watch the video: [link]. Take good notes, you will submit them as the first part of your homework.
• The homework problems are as follows. 
  
• Submit a copy of your notes for this section.
• §5.1 3, 5, 8, 17, 18, 19, 28, 32
• Note: 32 is a computational probem. Use CoLab's Python instead of MATLAB. The commands for generating random 3x3 matrices are np.random.rand(3,3) and np.random.randn(3,3). The first picks entries uniormly between 0 and 1; the second uses a normal distribution.
• After working on the problems by yourself, check in with your group. Discuss the material and the problems. Afterwards, designate someone to send me an update on your progress.  What was the most challenging part of the assignment? What parts of the lectures are still unclear?
• You should submit your homework solutions as a single .pdf file; a nice way to do create this document is to use a [phone scanner app]. This part of the homework is due on Thursday at 12pm. Submit it via [GradeScope].


Part 2


• Read section 5.2 or watch the video: [link]. Take good notes, you will submit them as the first part of your homework.
• The homework problems are as follows:
• Submit a copy of your notes for this section.
• §5.2 1, 5, 7, 9, 15, 27, 32*, 35*
• After working on the problems by yourself, check in with your group. Discuss the material and the problems. Afterwards, designate someone to send me an update on your progress.  What was the most challenging part of the assignment? What parts of the lectures are still unclear?
• You should submit your solutions as a single .pdf file; a nice way to do create this document is to use a [phone scanner app]. Homework is due on Monday at 12pm. Submit it via [GradeScope].:

In my lecture this week, I summarized the main goals of each of the sections of this week's reading. The slides for your reference are here: [PDF]. We will meet by Zoom at 1pm on Monday and I will hold an office hour on Thursday at 4pm.  Here are the links:

[Monday 1pm]

[Thursday 4pm]

If you would like to meet at any other point, please send me an email.

Assignment: There are two parts this week.  The first covers Section 5.3 and is to be completed by Thursday at noon. The second covers Section 6.1 and is to be completed by Monday at noon.

• Read section 5.3 or watch the video: [link]. Take good notes, you will submit them as the first part of your homework.
• The homework problems are as follows. 
  
• Submit a copy of your notes for this section.
• §5.3 1, 6, 12, 19, 20, 21, 28, 39*
• After working on the problems by yourself, check in with your group. Discuss the material and the problems. Afterwards, designate someone to send me an update on your progress.  What was the most challenging part of the assignment? What parts of the lectures are still unclear?
• You should submit your homework solutions as a single .pdf file; a nice way to do create this document is to use a [phone scanner app]. This part of the homework is due on Thursday at 12pm. Submit it via [GradeScope].


Part 2


• Read section 6.1 or watch the video: [link]. Take good notes, you will submit them as the first part of your homework.
• The homework problems are as follows:
• Submit a copy of your notes for this section.
• §6.1 3, 4, 9, 16, 19, 21, 22, 25, 33, 36*
• After working on the problems by yourself, check in with your group. Discuss the material and the problems. Afterwards, designate someone to send me an update on your progress.  What was the most challenging part of the assignment? What parts of the lectures are still unclear?
• You should submit your solutions as a single .pdf file; a nice way to do create this document is to use a [phone scanner app]. Homework is due on Monday at 12pm. Submit it via [GradeScope].:

In my lecture this week, I summarized the main goals of each of the sections of this week's reading. The slides for your reference are here: [PDF]. We will meet by Zoom at 1pm on Monday and I will hold an office hour on Thursday at 4pm.  Here are the links:

[Monday 1pm]

[Thursday 4pm]

If you would like to meet at any other point, please send me an email.

Assignment: There are two parts this week.  The first covers Section 6.2 and is to be completed by Thursday at noon. The second covers Section 6.4 and is to be completed by Monday at noon.

• Read section 6.2 or watch the video: [link]. Take good notes, you will submit them as the first part of your homework.
• The homework problems are as follows. 
  
• Submit a copy of your notes for this section.
• §6.2 1, 16, 20, 21, 26, 27, 28, 30, 32, 33, 36*
• After working on the problems by yourself, check in with your group. Discuss the material and the problems. Afterwards, designate someone to send me an update on your progress.  What was the most challenging part of the assignment? What parts of the lectures are still unclear?
• You should submit your homework solutions as a single .pdf file; a nice way to do create this document is to use a [phone scanner app]. This part of the homework is due on Thursday at 12pm. Submit it via [GradeScope].


Part 2


• Read section 6.4 or watch the video: [link]. Take good notes, you will submit them as the first part of your homework.
• The homework problems are as follows:
• Submit a copy of your notes for this section.
• §6.4 1, 3, 8, 10, 14, 18, 21, 26, 31*
• After working on the problems by yourself, check in with your group. Discuss the material and the problems. Afterwards, designate someone to send me an update on your progress.  What was the most challenging part of the assignment? What parts of the lectures are still unclear?
• You should submit your solutions as a single .pdf file; a nice way to do create this document is to use a [phone scanner app]. Homework is due on Monday at 12pm. Submit it via [GradeScope].:

The computational aspect of this class will be done in Google's Colaboratory. By way of an introduction, please read Google's own blurb about it: [link]. It is a wonderful platform to get your feet wet doing machine learning using Python.  We will use it as the computational platform for our linear algebra course.

You will need a Google account.  Computations occur in a notebook which can be simply saved as a Google Doc.  Start by working through the following notebook which illustrates how do use Python for some basic computations in linear algebra:

You will need to save your own copy of the notebook obtained from the following link:  go to File and then Save a copy to Drive.   Modify it as you see fit!

Introduction to matrix computations

Some of you may choose to complete a project that shows your mastery of some of the concepts covered in the course as well as exploring new topics in linear algebra.  You will hand in a written report or do a final presentation of your work, the goal of which is to show

• your mastery of knowledge of some component of linear algebra that we have studied this semester, as well as

• your ability to build upon what we have learned, either by learning additional material beyond what we have covered or by applying techniques of linear algebra to pursue some further goal.

The following tabs suggest some projects you can pursue. However, you should not feel limited to choose only from the projects presented here, nor should you view the project descriptions as instructions that must be followed to the letter. This is your chance to follow an interest in linear algebra!

Instructions

The projects will be due before Thanksgiving break.  You may work on these projects as part of a larger group, in which case your final paper or presentation should include substantial contribution from all the team members.  You are encouraged to use any external resources that you deem appropriate (both animate as well as inanimate), but you must make sure to cite them.

A reasonable written project is expected to be in the neighborhood of ten single-spaced pages.  Written work should demonstrate your mastery of the theoretical material as well as your ability to apply it to solve a problem.  In particular, a significant fraction of your written work should include an exposition of the relevant theory that we have learned in this course, and in certain cases, that you have learned from other sources during your research.
A presentation should last roughly thirty minutes and aim to satisfy the same goals as a written report.

As a general rule of thumb, you should aim to write a paper, or give a presentation, that could be read and understood by an outstanding student who has just completed a first course in linear algebra.

This project is based on one of my labs for a second course in linear algebra. The basic idea is encapsulated in the original lab document: [PDF]. The following CoLab notebook introduces the necessary computational tools:

Sound compression using the Haar basis

Project: Before you are ready for the fun stuff above, you will have to become an expert in changing the basis of a vector. In Strang's book, you will need to read Sections 7.1 and 7.2, and more formally, a Chapter 4 of Meyer's Matrix Analysis. Your project will be to describe how the change of basis procedure is used in sound compression. I am happy to leave the details open-ended, but at the end of the project, you should be comfortable with the theory of changing bases as well as be able to compare different sound compression schemes working on real songs and sounds.

This project is based on one of my labs for a second course in linear algebra. The basic idea is encapsulated in the original lab document: [PDF]. I also wrote a short CoLab notebook with some useful commands and a bare-bones outline of a sports-based version the project:

Ranking and the Perron-Frobenius Theorem.

Project: The first step is to figure out how the PageRank algorithm works. There are a number of resources online, but here are two: to get you started: [PDF], [PDF]. For this project, I would like you to describe the theory behind the PageRank algorithm, and then apply it to a ranking project of your own. There are many possibilities for the latter: using this idea for sports is discussed in the notebook above, but the idea has been used to rank super-spreader in disease propagation, importance of cities in road networks, squares in Monopoly, and many others.

Unlike the prior two projects, this one is more theoretical. Linear algebra, although originally formulated to deal with finite-dimensional vector spaces, has subsequently been used to work in infinite-dimensional settings as well. In fact, even though they did not have the language to describe it, mathematicians have been working with infinite-dimensional vector spaces for centuries.

• Start by learning about Fourier series from Strang's section 8.5 and then look for other resources online. This is a broad a beautiful field of mathematics with many available references.

In the first part of this project,write an introduction to Fourier analysis giving formulas for the Fourier coefficients of a function . Prove that the trigonometric functions are an orthonormal set. Explore what it might mean to span an infinite-dimensional vector space.

In the second part, explore the quality of these approximations for a variety functions. The functions are specified on the interval.

• 

• 

• 

Graph each function together with a few Fourier series approximations. Are they any good? Finally, read about the Gibb's phenomenon and the trouble it causes for the theory of Fourier series.