Course Information

Welcome to MATH 416: Abstract Linear Algebra! Linear algebra is an essential component of modern mathematics, physics, and computer science. This course will provide a rigorous treatment of the subject, focusing on the structure and beauty of vector spaces, linear maps, eigenvectors, and much more. In addition to being elegant and powerful, the course content will also provide ample opportunity for us to understand what mathematics is, what mathematicians do, and how they actually do it! Our official syllabus can be downloaded here; however, I've included some of the most important points below for easy reference.

• Instructor: Jacob Beckey
• Grader: Hyun Tark
• Meeting times: Mon, Wed, Fri — 1:00–1:50 PM, 136 Davenport Hall
• Office Hours: Mon 2-2:50 (214 Davenport Hall), Wed 2-2:50 (132 Davenport Hall), or by appointment

Contact & Communication

• Instructor's email: jbeckey@illinois.edu
• My office: 72 Computing Applications Building (hopefully changing to Harker Hall soon!)
• Grader's email: htark2@illinois.edu

Please reach out if you have questions about course content, assignments, or logistics.

Schedule & Grading

My goal in this course is to help you deeply learn linear algebra while also developing your ability to think abstractly, write cogent proofs, and work collaboratively. Education research has made clear that active engagement increases student learning outcomes and, even more importantly, mathematics is a fundamentally social endeavor. As such, class participation is a significant part of your grade. This grade will be determined based on participation, not correctness, of in-class clicker questions and pre-lecture quizzes. To accomodate unforeseen circumstances, I will drop the lowest two homework grades, the lowest midterm grade, and 5 participation items. With those caveats, your grade will be calculated based on the following breakdown.

Grading Breakdown

• Participation: 20%
• Homework: 30%
• Midterm Exams: 30%
• Final Exam: 20%

Homework

Problem sets will be assigned on Fridays and will be due (via Canvas) on the following Friday. LaTeX is ubiquitous in STEM research fields, so I implore all students to use it for writing up solutions, but this is not a requirement. If you have not ever used LaTeX, I recommend starting with this tutorial.

Important Dates

• Midterm Exam 1: Friday, September 26
• Midterm Exam 2: Friday, October 24
• Midterm Exam 3: Wednesday, November 19
• Final Exam: Wednesday, December 17 (7-10pm, location TBD)

Course Materials

Required Materials

• Textbook: Axler, Linear Algebra Done Right
• Technology: iClicker app (see student instructions here)

Suggested Resources

• Book: Vector: A Surprising Story of Space, Time, and Mathematical Transformation by Robyn Arianhod
• Video series: Essence of Linear Algebra by Grant Sanderson (3Blue1Brown on YouTube)

Course Outline

We will aim to cover most of the content in chapters 1, 2, 3, 5, 6, 7, and 8 of Axler. This is a lot of content; however, if you work hard in this course, its hard to overstate how useful it will be in your future. The core topic list will be as follows.

• Solving systems of linear equations
• Complex numbers, vector spaces, subspaces
• Span and linear independence, bases, dimension of vector spaces
• Linear maps, null space and range, matrices, isomorphic vector spaces, isomorphism theorem, dual spaces
• Invariant subspaces, eigenvalues, eigenvectors, upper-triangular matrices
• Inner products, norms, orthonormal bases, orthogonal complements
• Operators on inner product spaces, self-adjoint and normal operators, spectral theorem, isometries, singular value decomposition
• Generalized eigenspaces, multiplicity of eigenvalues, characteristic and minimal polynomial, Jordan normal form
• Trace and determinant

Weekly Outline