CAPP 30271: Mathematics for Computer Science and Data Analysis (Winter 2023)

General information

Instructor

Timothy Ng (timng@uchicago.edu)

Lecture meetings

Sec. 1: MWF 10:30–11:30, 1155 140C

Sec. 2: MWF 1:30–2:30, RY 276

Tutorials

1L01: F 2:30–3:30, 1155 140C

1L02: F 3:30–4:30, C 303

2L01: F 2:30–3:30, RY 277

2L02: F 3:30–4:30, RY 277

Links

Overview

This course is an introduction to linear algebra with a slight emphasis on applications to data science. Traditionally, linear algebra is the study of linear systems. Such systems are described only by using scaling and addition. A key observation is that a table of data can also be viewed as a linear system. One can then exploit the structural properties of linear systems to answer questions about data. Linear algebra gives us the tools to do so.

Topics

Introduction to linear algebra: Vectors, matrices, linear combination
Matrices as linear systems: Elimination, permutation, LU factorization
Vector spaces: Fundamental subspaces, linear independence, basis, dimensionality
Orthogonality: Orthogonality, projections, Gram-Schmidt, least squares
Eigenvectors: Determinants, eigenvalues, diagonalization
The singular value decomposition: Singular values, principal component analysis

Text and resources

The textbook for this course is Gilbert Strang, Linear Algebra for Everyone (2020). Lecture notes for each class will also be made available following each class.

Linear algebra is a very popular subject nowadays and there are a wealth of resources available. Here are a few suggestions if you are seeking additional resources.

The following are video resources:
- Gilbert Strang’s recorded lectures from MIT’s 18.06 on MIT OpenCourseWare.
- Grant Sanderson’s series on 3Blue1Brown.
Other textbooks include:
- Introduction to Linear Algebra, 5th ed. by Gilbert Strang. This is an older but more comprehensive version of the text for this course.
- Linear Algebra Done Wrong by Sergei Treil. This gives a more formal mathematical treatment of linear algebra.
- Matrix Methods in Data Mining and Pattern Recognition by Lars Eldén. This is a more advanced text with more focus on applications to data science. It is freely available via the UChicago Library.

Communication

There are a number of different tools we’ll be using to communicate about the class.

Course materials. Lecture notes will be made available via this course website. The course webpage will also contain course information and can be treated as the syllabus.
Discussion and announcements. We will use Ed Discussion, an online discussion forum, for course discussion and announcements. Restricted course materials will also be posted here. You should make it a habit to check Ed for updates and announcements regularly.
Coursework. All coursework, including homework and tests, will be distributed and submitted via PrairieLearn. It is important that you have access to a device with an internet connection and a web browser during tests in class. Instructions for accessing the PrairieLearn course instance will be given in class.

Evaluation

Your computed grade in this course will be determined by the following coursework components.

Weekly homework, worth 25% in total.
Quizzes, worth 25% in total.
A midterm examination and a final examination, worth 50% in total.
- The midterm examination grade will be scaled to $\frac 1 4$.
- The final examination will then be worth the remainder of the 50%.

Homework

Homework is assigned and to be completed online on PrairieLearn. Homework is intended to give you practice for the tests. You should attempt to complete it on your own as much as possible.

Homework will be released on Wednesdays and should be completed by the next Friday following (10 days).

Many homework exercises are required to be done multiple times in order to get full credit. Such questions have a value, a point total, and a point maximum.

If you answer a question correctly, two things happen:
- The point total increases by the value, until you reach the point maximum.
- The value increases (2x, 3x, and so on).
If you answer a question incorrectly, one thing happens:
- The value goes back to what it was originally (1x).

What does this mean? If you can answer a question correctly multiple times in a row, it shows that you've likely learned the underlying idea. If you answer a question incorrectly, there is no penalty, but you will need to answer the question correctly more times (i.e. practice). Note that your score will never decrease.

If you have already reached the maximum number of points, you can continue to answer questions to get more practice—very useful, since you can always generate new questions and have them graded immediately.

Tests

Quizzes and examinations are taken in-person, on PrairieLearn. You will require access to a device with an internet connection and a web browser during test sessions.

Quizzes will be taken during tutorials each week. Quizzes will be 15–20 minutes.
The midterm examination will be held during tutorial of fifth week instead of a quiz. The midterm examination will be 50 minutes—the length of the entire tutorial session.
The final examination will be held in the scheduled final examination time slot. The final examination will be two hours.

Lectures

Section numbers refer to Strang (LAFE).

January 4: Introduction, vectors (1.1)
January 6: Geometry, linear combinations, matrices (1.1–1.3)
January 9: Linear independence, matrix products and decomposition (1.3–1.4)
January 11: Matrices as systems of linear equations, matrices as operations (1.4–2.1)
January 13: Elimination, inverses (2.1–2.2)
January 18: LU factorization (2.2–2.3)
January 20: Permutations, inverses (2.3–2.4)
January 23: Vector spaces (3.1)
January 25: The null space (3.2)
January 27: Properties from the RREF(3.2–3.3)
January 30: The complete solution to $A \mathbf x = \mathbf b$ (3.3)
February 1: Basis (3.4)
February 3: Dimension (3.4)
February 6: Fundamental subspaces of a matrix (3.5)
February 8: Orthogonality of the fundamental subspaces (4.1)
February 10: Projections (4.2)
February 13: Least squares (4.2–4.3)
February 15: Orthogonal matrices and the Gram–Schmidt process(4.4)
February 17: QR factorization, eigenvalues and eigenvectors (4.4, 6.1)
February 20: Computing eigenvalues and eigenvectors (6.1, 5.1–2)
February 22: Diagonalization (6.2)
February 24: Symmetric matrices (6.3)
February 27: The singular value decomposition (7.1)
March 1: Low-rank approximations (7.3)
March 3: Epilogue, Q&A

Academic integrity

It is your responsibility to be familiar with the University’s policy on academic honesty. Instances of academic dishonesty will be referred to the Office of the Provost for adjudication. Following the guidelines above on collaboration and citation should be sufficient, but if you have any questions, please ask me (the instructor).

Accessibility

Students with disabilities who have been approved for the use of academic accommodations by Student Disability Services (SDS) and need reasonable accommodation to participate fully in this course should follow the procedures established by SDS for using accommodations. Timely notifications are required in order to ensure that your accommodations can be implemented. Please meet with me (the instructor) to discuss your access needs in this class after you have completed the SDS procedures for requesting accommodations.