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# MATH 221: Linear Algebra

### 2020 Fall, Emory University

#### Contacts

 Lecture Instructor Dr. Le Chen Email le.chen@emory.edu (please include "Math 221" in the subject field) Office Math & Science Center - E431 Office hours Tue, 4:00pm -- 5:00pm, Thur, 2:30pm -- 3:30pm, or by appointments. Zoom Classroom Click here (password on Canvas page).
Sections Class meeting Lab (Friday) Lab Instructor
Math221-5 Tue: 1:00PM - 2:15PM 11:20AM - 12:10PM Zitong Pei
Math221-6 Thu: 1:00PM - 2:15PM 1:00PM - 1:50PM Michael Cerchia
Math221-7 Tue: 2:40PM - 3:55PM 2:40PM - 3:30PM Zitong Pei
• Lectures will be delivered on Tuesday and Thursday. You need to connect to the Zoom class
meeting through the link below (password can be found on Canvas page):
• The same link will be used for office hours.
• On Friday, the lab session will be host by the lab instructors:

#### Course description

(Image from Wikipedia)

Linear algebra is a branch of mathematics concerning linear equations such as $$a_1 x_1 + \cdots + a_n x_n = b$$, linear maps such as $$(x_1,\ldots,x_n) \mapsto a_1x_1 + \cdots + a_n x_n$$, and their representations in vector spaces and through matrices. Linear algebra is central to almost all areas of mathematics. It is also used in most sciences and engineering areas.

This course begins with the definition of a matrix and some fundamental operations that can be performed on matrices, such as adding or multiplying two matrices together. Vector spaces are also introduced. A connection is then formed by modeling vector spaces using matrices. Advanced topics involving matrices, such as diagonalization and quadratic forms, eigenvalues and eigenvectors, orthogonalization, and the Gram-Schmidt process are examined. At the end of the class, you will understand how google ranks web pages.

#### Textbook

We will use the following online book which is free to download:

#### Coverage

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

• Chapter 1. Systems of Linear Equations.
• Chapter 2. Matrix Algebra
• Chapter 3. Determinants and Diagonalization
• Chapter 4. Vector Geometry
• Chapter 5. Vector Space $$\mathbb{R}^n$$
• Chapter 6. Vector Spaces
• Chapter 7. Linear Transformations
• Chapter 8. Orthogonality
• Chapter 9. Change of Basis
• Chapter 10. Inner Product Spaces
• Chapter 11. Canonical Forms

#### Prerequisite

• Six hours of AP BC credit, or
• Math 12 or 112Z

#### Learning outcomes

By the end of this course, students will be able to

• Solve system of linear equations, perform matrix operations, find the inverse of a matrix, check
vectors for linear independence.
• List the vector space properties for $$\mathbb{R}^n$$, give examples of subspaces, find bases for
subspaces, find its dimension, find orthogonal bases for subspaces, identify linear
transformations.
• Find eigenvalues and eigenvectors, determinants, characteristic polynomial, perform
diagonalization of a matrix.
• Give examples of vector spaces, check for linear independence, find bases, dimension, identify
linear transformations and find matrix representation for linear transformations.

#### Students obligations

In order to successfully master the material and complete the course, you are expected to

• Read the textbook and watch video lectures for each section.
• Attend weekly Zoom class meeting and weekly lab session, and watch video recordings for the other weekly class meeting.
• Response to the Zoom poll during the class meeting.
• Participate actively in group activities during Zoom meetings.
• Complete and submit weekly homework through gradescope.
• Read solutions and any feedback you receive for each problem set.
• Complete three midterm tests and one final exam.
• Use appropriate etiquette and treat other students with respect in all discussions.
• Do not hesitate to ask for help whenever needed.

Note: The syllabus was created in May 2020, and it is subject to changes during the semester.

#### Homework

• There will be eleven weekly homework assignments.
• Homework of a week will be usually assigned by the Friday of that week.
• It will be due on the following week on Thursday at 1pm unless announced otherwise.
• No late homework will be accepted.
• The homework consists of two phases:
• In Phase I, you need to complete questions on Canvas Quiz.
• In Phase II, you need to write details of some problems and upload your solutions to gradescope.
• Keep in mind that, the Canvas Quiz part, no matter it is for homework, test or final exam, you
will be given two attempts and the latest attempt will be recorded.
• The lowest grade will be dropped, that is, the final score for the homework will be averaged over the rests.
• Note that the drop policy is not a bonus. It aims at accounting for all circumstances
such as sickness, injuries, family emergencies, religion holidays, etc.

#### Midterm tests

• There will be three midterm tests in the following three Fridays.
Date
Test I 09/04/2020
Test II 10/02/2020
Test III 10/30/2020
• Please note down the above three dates. There will be no late exam. In case of time
• The test consists of two phases as the homework. The only difference is that you need to complete
both Phases in 24 hours.

#### Due dates for homework and tests

• Here are a list of due dates for eleven homeworks and three tests (the Phase II part).
Tuesday   Thursday   Friday   Saturday
Week 1     08/19   08/21
Week 2 08/25   08/27 HW01 due at 1pm 08/28
Week 3 09/01   09/03 HW02 due at 1pm; RS 09/04 Test I 09/05 Test I due at 6pm
Week 4 09/08   09/10   09/11
Week 5 09/15   09/17 HW04 due at 1pm 09/18
Week 6 09/22   09/24 HW05 due at 1pm 09/25
Week 7 09/29   10/01 HW06 due at 1pm; RS 10/02 Test II 10/03 Test II due at 6pm
Week 8 10/06   10/08   10/09
Week 9 10/13   10/15 HW08 due at 1pm 10/16
Week 10 10/20   10/22 HW09 due at 1pm 10/23
Week 11 10/27   10/29 HW10 due at 1pm; RS 10/30 Test III 10/31 Test III due at 6pm
Week 12 11/03   11/05   11/06
Week 13 11/10   11/12 HW11 due at 1pm 11/13
Week 14 11/17   11/19 RS 11/20
Week 15 11/24 HW12 due at 1pm; RS
• RS: review session.

#### Final exam

• Final exam will be cumulative.
• It will be in a similar form as the midterm tests, namely, you will have two phases:
• Phase I: Canvas quiz at the following time slot:
• Tuesday, Dec. 8th, from 6:30-9:00pm (Atlanta local time)
• There will be 75 points in total with 16 choice problems and 5 points each. One problem is
bonus problem.
• It will be released on Dec. 8th at 9:00pm
• It will be due on Dec. 9th at 9:00pm
• The total points are 25 points.
• There will be about two proof-based problems.

#### Attendance

• The attendance will be collected during Zoom Sessions via Zoom Poll.
• You will need to answer questions during the Zoom sessions actively in order to count your attendance.
• either at the beginning of the semester for the whole semester;
• or at least one day before the class meeting for a specific Zoom session.
• The attendance points will be computed as follows
$\text{Attendance points} = 100 \times \text{percentage of polls that you respond throughout the semester.}$

#### Assessment

• The final score will be determined as follows:
• Based on the final score (plus potential bonus points), the final letter grade will be determined as follows:
A 92%-100% A- 90%-91.9%
B+ 87%-89.9% B 82%-86.9% B- 80%-81.9%
C+ 77%-87.9% C 72%-76.9% C- 70%-71.9%
D+ 67%-67.9% D 67%-67.9% D- 60%-61.9%
F 0%-59.9%
• Curving policies: I may make a curve of the grades after the final exam as I have promised. Here
is the slight more detail that how I would do that: You are eligible for grade curving provided
that you did better in the final than the semester average. So do not give up and take the final
exam as a rare opportunity to solidify your knowledge of linear algebra.

#### Text and slides

• Slides will be updated constantly throughout the semester and please check the time stamp on the front page.
• The application parts of each chapter will be skipped due to lack of time. However, the materials are given for motivated students to study.
Chapter/Section Text Slides Slides
Chapter 1. Systems of Linear Equations text
Section 1-1. Solutions and Elementary Operations text presentation handout
Section 1-2. Gaussian Elimination text presentation handout
Section 1-3. Homogeneous Equations text presentation handout
Section 1-4. An Application to Network Flow text
Section 1-5. An Application to Electrical Network text presentation handout
Section 1-6. An Application to Chemical Reactions text presentation handout
Section 1-S. Supplementary Exercises for Chapter 1 text
Chapter 2. Matrix Algebra text
Section 2-1. Matrix Addition and Scalar Multiplication and Transportation text presentation handout
Section 2-2. Matrix-Vector Multiplication text presentation handout
Section 2-3. Matrix Multiplication text presentation handout
Section 2-4. Matrix Inverses text presentation handout
Section 2-5. Elementary Matrices text presentation handout
Section 2-6. Linear Transformations text presentation handout
Section 2-7. LU-Factorization text presentation handout
Section 2-8. An Application to Input-Output Economic Model text
Section 2-9. An application to Markov Chains text presentation handout
Section 2-S. Supplementary Exercises for Chapter 2 text
Chapter 3. Determinants and Diagonalization text
Section 3-1. The Cofactor Expansion text presentation handout
Section 3-2. Determinants and Matrix Inverses text presentation handout
Section 3-3. Diagonalization and Eigenvalues text presentation handout
Section 3-4. An Application to Linear Recurrences text presentation handout
Section 3-5. An Application to Systems of Differential Euqations text
Section 3-6. Proof of the Cofactor Expansion Theorem text
Section 3-S. Supplementary Exercises for Chapter 3 text
Chapter 4. Vector Geometry text
Section 4-1. Vectors and Lines text presentation handout
Section 4-2. Projections and Planes text presentation handout
Section 4-3. More on the Cross Product text presentation handout
Section 4-4. Linear Operators on R3 text presentation handout
Section 4-5. An Application to Computer Graphics text
Section 4-S. Supplementary Exercises for Chapter 4 text
Chapter 5. Vector Space Rn text
Section 5-1. Subspaces and Spanning text presentation handout
Section 5-2. Independent and Dimension text presentation handout
Section 5-3. Orthogonality text presentation handout
Section 5-4. Rank of a Matrix text presentation handout
Section 5-5. Similarity and Diagonalization text presentation handout
Section 5-6. Best Approximation and Least Squares text
Section 5-7. An Application to Correlation and Variance text
Section 5-S. Supplementary Exercises for Chapter 5 text
Chapter 6. Vector Spaces text
Section 6-1. Examples and Basic Properties text presentation handout
Section 6-2. Subspaces and Spanning Sets text presentation handout
Section 6-3. Linear Independence and Dimensions text presentation handout
Section 6-4. Finite Dimensional Spaces text presentation handout
Section 6-5. An Application to Polynomials text
Section 6-6. An Application to Differential Equations text
Section 6-S. Supplementary Exercises for Chapter 6 text
Chapter 7. Linear Transformations text
Section 7-1. Examples and Elementary Properties text presentation handout
Section 7-2. Kernel and Image of a Linear Transformation text presentation handout
Section 7-3. Isomorphisms and Composition text presentation handout
Section 7-4. A Theorem about Differential Equations text
Section 7-5. More on Linear Recurrences text
Chapter 8. Orthogonality text
Section 8-1. Orthogonal Complements and Projections text presentation handout
Section 8-2. Orthogonal Diagonalization text presentation handout
Section 8-3. Positive Definite Matrices text presentation handout
Section 8-4. QR-Factorization text presentation handout
Section 8-5. Computing Eigenvalues text presentation handout
Section 8-6. The Singular Value Decomposition text presentation handout
Section 8-7. Complex Matrices text
Section 8-8. An Application to Linear Codes over Finite Fields text
Section 8-9. An Application to Quadratic Forms text
Section 8-10. An Application to Constrained Optimization text
Section 8-11. An Application to Statistical Principal Component Analysis text

Here is the list of all lecture videos for the whole semester.

#### Tentative schedule

• Below is the tentative schedule that may change during the semester:
Tuesday (lecture)   Thursday (lecture)   Friday (lab)
Week 1     08/19 First class, 1.1 08/21
Week 2 08/25 1.2-1.3 08/27 2.1-2.2 08/28
Week 3 09/01 2.2 - 2.3 09/03 RS 09/04 Test I
Week 4 09/08 2.4 09/10 2.5 09/11
Week 5 09/15 2.6-2.7 09/17 3.1 09/18
Week 6 09/22 3.2 09/24 3.3 09/25
Week 7 09/29 RS 10/01 RS 10/02 Test II
Week 8 10/06 4.1 10/08 4.2 10/09 4.3
Week 9 10/13 5.1 10/15 5.2 10/16
Week 10 10/20 5.3-5.4 10/22 5.5 10/23
Week 11 10/27 RS 10/29 RS 10/30 Test III
Week 12 11/03 6.1-6.2 11/05 6.3(-6.4) 11/06
Week 13 11/10 8.1-8.2 11/12 8.3-8.4 11/13
Week 14 11/17 8.6 11/19 RS 11/20
Week 15 11/24 Last class (RS)
• RS: Review Session
• For students who take synchronized session on Tuesday (resp. Thursday)
• Option 1 (recommended): please timely watch recording for Thursday (resp. Tuesday), which should be available in the same day.
• Option 2 (acceptable): you can also attend the session(s) on Thursday (resp. Tuesday).

• We will use gradescope to handle submissions of homework, take-home tests and exams,
which allows us to provide fast and accurate feedback on your work.
• As soon as grades are posted, you will be notified immediately so that you can log in and see your
• You should have received an email from Gradescope for the registration (sent out on 07/23/2020);
• If you have not yet received this email, please use the Entry Code to register yourself: 9Y6R48.
• Printer+scanner or tablet
• The easiest way to submit the homework/tests/exams is the following steps:
1. print the given template;
2. complete the problem sets;
3. scan the resulting paper (make sure it is legible);
• Alternatively, if you have a tablet that you can write on it, you may simply write on the
template pdf file and upload the resulting file.
• The following short video (1 minutes 40 seconds) shows the basic usage of gradescope, which should
explain everything you need to be able to do.

• More instruction will be available towards the fall 2020.

#### Feedback

• Your feedbacks are important for us to improve the teaching and make the learning process more effective and enjoyable.
• Here are two ways that you could let me know what your think:
1. You may send me an email.
2. If you want to send me some feedback in an anonymous way, you may fill in the following form:

#### Netiquette

Not all forms of communication found online are appropriate for an academic community or respectful of others. In this course (and in your professional life that follows), you should practice appropriate etiquette online (netiquette''). Here are some guidelines:

• You should read and follow Rasmussen College's 10 Netiquette guidelines every online students needs to know.
• You are encouraged to login Zoom in time (preferably one or two minutes early) and stay until the end.
• During the zoom class session,
• please silence all cell phones and other electronic devices;
• Please do not read email or look at websites, social media, etc;
• please actively participate in the zoom class meeting:
• You should also expect that I randomly pick students to help me solve exercises
• There will be Zoom Poll from time to time:
You need to response the questions during Zoom Sessions to justify your active attendance.

#### Honor code

• Students should familiarize themselves with the Emory College honor code here
• Students are encouraged to share ideas and solutions on problem sets and labs, but must
express those ideas in their own words in their submitted work.
• Students are not authorized to view or use the work of another student during exams.

#### Accessibility

Your success in this class is important to me. We will all need accommodations because we all learn differently. If there are aspects of this course that prevent you from learning or exclude you, please let me know as soon as possible. Together we’ll develop strategies to meet both your needs and the requirements of the course.

I encourage you to visit the Office of Accessibility Services (OAS) to determine how you could improve your learning as well. You can register and make a request for services from OAS. In this case, please do inform me of such requests. See the following link for more information:

#### Harassment

Discriminatory harassment of any kind, whether it is sexual harassment or harassment on
the basis of race, color, religion, ethnic or national origin, gender, genetic
information, age, disability, sexual orientation, gender identity, gender expression,
veteran’s status, or any factor that is a prohibited consideration under applicable law,
by any member of the faculty, staff, administration, student body, a vendor, a contractor,
guest or patron on campus, is prohibited at Emory.


#### Acknowledgement

• This syllabus is created following some sample syllabuses from my colleagues which
include Mike Carr, Dwight Duffus, Bree Ettinger, Manuela Manetta, James Nagy.