# Linear Algebra (MATH-UA 140, NYU)

Undergraduate course, *NYU Courant*, 2024

**Syllabus**: pdf

- Formulate, solve, apply, and interpret systems of linear equations in several variables;
- Compute with and classify matrices;
- Master the fundamental concepts of abstract vector spaces;
- Decompose linear transformations and analyze their spectra (eigenvectors and eigenvalues);
- Utilize length and orthogonality in each of the above contexts;
- Apply orthogonal projection to optimization (least-squares) problems;
- Explore other topics (as time permits).

# Logistics

**Time and Location**: Monday and Wednesday

- 9:30 - 10:45 at RGSH 203 (Brooklyn) (Midterm: 3/13/2024 in class, Final: Mon 5/13/2024 8:00AM - 10:00AM Tandon, Jacobs Academic Bldg Room 475)
- 3:30 - 4:45 at 19 West 4th Street 101 (Washington Square) (Midterm: 3/13/2024 in class, Final: Fri 05/10/2024 12:00pm-14:00pm Cantor Film Center, Room 102)

**Office Hour**:

- 10:45 - 11:45 Monday, Wednsday room 870, 2 MTC
- 2:00 - 3:00 Wednsday campuswire, Course code see brightspace

**TA Office Hour:**

- 2:00 - 3:00 Monday, campuswire, by Ao
- 3:00 - 5:00 Thursday, 524 WWH, by Adithya
- 9:30 AM - 10:30 AM Thusday and 12 PM - 1 PM Friday, 228 WWH 228, by Animesh

**ChatGPT Tutor**: The link here provides a Large Language Model agents that is specifically trained for this course.

If you have a question, to get a response from the teaching staff quickly we strongly encourage you to post it to the class Piazza forum. For suggestions to improve Yiping’s teaching, you can use the (anonymous) form. You can also leave a private matter here. If you wish to contact me via email, kindly include the tag “[Linear Algebra]” in the subject line. This will help ensure that I do not overlook your message.

For longer discussions with TAs and to get help in person, we strongly encourage you to come to office hours.

Free tutoring: Math Department, University Learning Center

**Course grades:** The final course grade will be based on the following breakdown:

- Attendance & Participation 5%
- Quizzes 15%
- Problem Sets 10%
- Exams 70% ( Midterm: 3/13, Final Exam: 5/10(CAS) 5/13(Tandon))

# Syllabus

Gilbert Strang’s course video: link

Cheat Sheet: [link], Practice Problem Sets :[link], Past Exams from MIT: [link]

- Lecture 1, Strang Sections 1.1 and 1.2 (Vector space, dot product) ,1/22 [slide] [annotated note]
- Lecture 2, Strang Sections 1.2 and 1.3 (Recap of vector space, Span), 1/24 [slide] [annotated note]
- Lecture 3, Strang Sections 2.1 and 2.2 (Matrices, Linear Systems), 1/29 [slide] [annotated note]
- Lecture 4, Strang Sections 2.1 and 2.2 (Linear Systems, Linear Dependence and Independence), 1/31 [slide] [annotated note]
- Lecture 5, Strang Sections 2.1 and 2.4 (Linear Systems, Matrix Operations), 2/5 [slide] [annotated note] [Gilbert Strang’s Video 1] [Gilbert Strang’s Video 3]
- Lecture 6, Strang Section 2.2 and 2.3 (Elimination, Elimination using Matrices), 2/7 [slide] [annotated note] [Gilbert Strang’s Video 2]
- Lecture 7, Review and Strang Section 2.5 (Review and Inverse Matrix), 2/11 [slide] [annotated note] [Gilbert Strang’s Video 3]
- Lecture 8, Strang Section 2.5 and 2.6 (Inverse matrix and LU Decomposition), 2/13 [slide] [annotated note] [Gilbert Strang’s Video 4]
- Lecture 9, No Class, 2/19,
- Lecture 10, Strang Section 2.6 (LU, LDU, LDL Decomposition), 2/21, [slide] [annotated note]
- Lecture 11,Strang Sections 2.7 and 3.1 (Spaces of Vectors, Nullspace), 2/26 [slide1] [annotated note] [Gilbert Strang’s Video 5]
- Lectrue 12, Strang Sections 3.2 and 3.3 (Nullspace, The Complete Solutions), 2/28 [slide1] [slide2] [annotated note] [Gilbert Strang’s Video 6] [Gilbert Strang’s Video 7] [Gilbert Strang’s Video 8]
- Lectrue 13, Strang Section 3.4 (Independence, Basis and Dimension) 3/4 [slide] [annotated note] [Gilbert Strang’s Video 9]
- Lectrue 14, Strang Section 3.5 (Orthogonality of the Four Subspaces, Review) 3/6 [slide] [Gilbert Strag’s Video 10]
- [Midterm cheat sheet] (Page 9 summarize the most important properties)
- [Midterm Review Problems] [Midterm Review Problems Answer] [Midterm Review Problems Filled]

- Lectrue 15, Strang Section 4.1 and 4.2 (Projection and Review) 3/11 [slide] [annotated note] [Gilbert Strag’s Video 14] [Gilbert Strag’s Video 15]
- Lectrue 16, Midterm, 3/13
- [Midterm cheat sheet] (Page 9 summarize the most important properties)
**Concept Questions**: [Review Question 1] [Sample Questions]**Calculation Questions**: [Compute Inverse of a Matrix] [Compute LU] [Compute Row space, Column space, Null space]- [Past Exams at NYU] [Past Exams from MIT]

- Lecture 17, Section 4.3 , Applications of Linear Algebra 1 (Linear Regression and Data Analysis), 3/25 [slide] [annotated note] [Gilbert Strag’s Video 16]
- Lecture 18, Section 4.4, Orthogonal basis and QR Decomposition, 3/27 [slide] [annotated note] [Gilbert Strag’s Video 17]
- Lecture 19, Section 5.1, Determinats, 4/1 [slide] [annotated note] [Gilbert Strag’s Video 18]
- Lecture 20, Section 5.3, Cofactors, 4/3 [slide] [annotated note] [Gilbert Strag’s Video 19] [Gilbert Strag’s Video 20]
- Lecture 21, Eigenvalues, 4/8 [slide] [annotated note] [Gilbert Strag’s Video 21]
- Lecture 22, Diagonalization , 4/10 [slide] [annotated note] [Gilbert Strag’s Video 22]
- Lecture 23, Symmetric Matrix, 4/15 [slide] [annotated note] [Gilbert Strag’s Video 25]
- Lecture 24,
*(NOT IN FINAL)*Applications of Linear Algebra 2 (Graph as matrix: Google Pagerank, Social Network Community Detection), 4/17 [Slide]- Yiping is out of town today, so we’ll record the lecture [Video]
- [Take Home Exerciese]
- [power methods]
- [Gilbert Strag’s Video 22] [Gilbert Strag’s Video 23] [Gilbert Strag’s Video 24]

- Lecture 25, Singular Value Decomposition , 4/22 [slide] [annotated note] [Gilbert Strag’s Video 29]
- Lecture 26, Review on Eigenvalue and SVD, 4/24 [notes version 1] [notes version 2] [Gilbert Strag’s Video 29]
- Review Exercise [pdf]

- Lecture 27, Linear Transform and Change of Basis , 4/29 [Note] [slide1] [slide2] [Gilbert Strag’s Video 30] [Gilbert Strag’s Video 31]
- Lecture 28, Final Review 5/1
- Final: [Review Note] [Review Questions] [Review Question Answers]
- Covers: note Projection, QR Decomposition Gram-schmidt, least squre, Determinates, Cofactors, Cramer’s Rule, diagnoalization (Eigendecomposition), Symmetric Matrix, SVD, finding the orthonormal basis of the four fundemental subspaces, Linear Transformation Check, Change of Basis.
- Past Exams: link
- Tandon Session: Mon 5/13/2024 8:00AM - 10:00AM Tandon, Jacobs Academic Bldg Room 475
- CAS Session: Fri 05/10/2024 12:00pm-14:00pm Cantor Film Center, Room 102

# Exams

- Midterm (3/13) Morning Session Afternoon Session Makeup Exam
- Final (05/10) Morning Session

**Additional reading (Not required for final)**

- Applications of Linear Algebra 3 (Unsupervised Data analysis (principal component analysis, Netflix competition, Video Analysis)) [Slide]
- [Simple Note for PCA] Another Note on PCA [Advanced Reading]

# ChatGPT

While the use of AI tools to aid in problem-solving is becoming increasingly prevalent, it is important to note that relying solely on AI to complete your homework is not in accordance with the expectations of this course. Submitting AI-generated solutions without proper acknowledgment is a violation of ethical guidelines and academic standards.

I’m collecting questions that Chatgpt/Bard provides wrong answers. If you find one, please report here in (anonymous) form.

# General Policies

In general, we do not grant extensions on assignments/exams. There are several exceptions:

*Medical Emergencies:*If you are sick and unable to complete an assignment or attend class, please go to University Health Services. For minor illnesses, we expect grace days or our late penalties to provide sufficient accommodation. For medical emergencies (e.g. prolonged hospitalization), students may request an extension afterward by contacting their Student Liaison or Academic Advisor and having them reach out to the instructor on their behalf. Please plan ahead if possible.*Family/Personal Emergencies:*If you have a family emergency (e.g. death in the family) or a personal emergency (e.g. mental health crisis), please contact your academic adviser or Counseling and Psychological Services (CaPS). In addition to offering support, they will reach out to the instructors for all your courses on your behalf to request an extension.*University-Approved Absences:*If you are attending an out-of-town university-approved event (e.g. multi-day athletic/academic trip organized by the university), you may request an extension for the duration of the trip. You must provide confirmation of your attendance, usually from a faculty or staff organizer of the event.

**Accommodations for Students with Disabilities:** If you have a disability and have an accommodation letter from the Disability Resources office, I encourage you to discuss your accommodations and needs with the Moses Center for Student Accessibility as early as possible in the semester for assistance. (Telephone: 212-998-4980, Website: http://www.nyu.edu/csd) We will work with you to ensure that accommodations are provided as appropriate. If you suspect that you may have a disability and would benefit from accommodations but are not yet registered with the Office of Disability Resources, I encourage you to contact them. Please note that it is your responsibility to schedule exams at the Moses Center, and to ensure that you are receiving all accommodations you are approved for.

**Collaboration among Students:** The purpose of student collaboration is to facilitate learning, not to circumvent it. Studying the material in groups is strongly encouraged. It is also allowed to seek help from other students in understanding the material needed to solve a particular homework problem, provided no written notes (including code) are shared, or are taken at that time, and provided learning is facilitated, not circumvented. The actual solution must be done by each student alone.

The presence or absence of any form of help or collaboration, whether given or received, must be explicitly stated and disclosed in full by all involved. Specifically, each assignment solution must include answering the following questions:

- Did you receive any help whatsoever from anyone in solving this assignment? Yes / No. If you answered ‘yes’, give full details:
**__****__**(e.g. “Jane Doe explained to me what is asked in Question 3.4”) - Did you give any help whatsoever to anyone in solving this assignment? Yes / No. If you answered ‘yes’, give full details:
**__****__**_ (e.g. “I pointed Joe Smith to section 2.3 since he didn’t know how to proceed with Question 2”)