Course information
Welcome to my class MATH 416 Abstract Linear Algebra! In this course we will study the structure of vector spaces and linear maps between vector spaces. Please see the Table of contents below for a detailed list of topics that we will cover.
Enrolled students: We will use Canvas for handing in homework and taking the quizzes before each lecture. Please make sure that you have access to the Canvas course website; otherwise, please contact me.
Lectures
The class follows the "flipped classroom" paradigm: I will make pre-recorded lectures of the course material available on this website. You are supposed to watch them at home, and take a short quiz about the lecture before class starts. In class we will answer your questions about the material and solve additional problems in group work.
Grading policy
- Class participation: 10%
- Quizzes: 10%
- Homework: 20%
- Midterms: 30%
- Final exam: 30%
In order to accommodate unforeseen circumstances that might prevent you from keeping up with the class, I will drop the three lowest homework scores, the three lowest quiz scores, and the lowest of the three midterm exam scores.
Homework
Homework will be assigned every Thursday of class (except November 25 and December 2), and has to be handed in via Canvas by the following Thursday.
Midterms and final exam
- Midterm 1: Friday, September 24
- Midterm 2: Friday, October 29
- Midterm 3: Friday, November 19
- Final exam: on or after Friday, December 10 (exact date TBD)
Code of conduct
I am dedicated to providing an inclusive and safe classroom experience for everyone, regardless of gender, gender identity and expression, sexual orientation, disability, physical appearance, body size, race, age or religion. I will not tolerate harassment and discriminating or disrespectful behavior between any classroom participants (including myself) in any form, whether in person or online. Violations of this code of conduct will be reported appropriately. (This code of conduct is based on a template provided by the Geek Feminism Wiki.)
A word of advice
The importance of linear algebra within mathematics cannot be overstated, as it serves as the foundation of many more advanced topics. In order to master the material, I highly recommend active participation in the class, which can be achieved by watching my recorded lectures, attending class, solving the assigned homework problems, and preparing for midterms and the final exam.
Finally, I'd like to stress that discussion and collaboration are absolutely essential parts of mathematics. I encourage you to discuss the material with fellow students and in class together with me. Please don't hesitate to visit me in my office hours or contact me via email if you have any questions about class or any other concerns!
Table of contents
- Solving systems of linear equations
- Fields, 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
Lectures
All lecture videos are collected in the channel MATH 416 Abstract Linear Algebra on the Illinois Media Space.
Week 1
- Lecture 1 (Mon, Aug 23): Introduction [recording] [lecture notes]
- Lecture 2 (Wed, Aug 25): Gaussian elimination [recording] [lecture notes]
- Lecture 3 (Fri, Aug 27): Geometry of linear systems [recording] [lecture notes]
- Homework: [sheet1]
Week 2
- Lecture 4 (Mon, Aug 30): Fields [recording] [lecture notes]
- Lecture 5 (Wed, Sep 1): Vector spaces [recording] [lecture notes]
- Lecture 6 (Fri, Sep 3): Subspaces and direct sums [recording] [lecture notes]
- Homework: [sheet2]
Week 3
- Lecture 7 (Wed, Sep 8): Span and linear independence [recording] [lecture notes]
- Lecture 8 (Fri, Sep 10): Bases [recording] [lecture notes]
- Homework: [sheet3]
Week 4
- Lecture 9 (Mon, Sep 13): Dimension of a vector space [recording] [lecture notes]
- Lecture 10 (Wed, Sep 15): Linear maps [recording] [lecture notes]
- Lecture 11 (Fri, Sep 17): Kernel and image of a linear map [recording] [lecture notes]
Week 5
- Lecture 12 (Mon, Sep 20): Fundamental theorem of linear algebra [recording] [lecture notes]
- Lecture 13 (Fri, Sep 24): Matrix representation of linear maps [recording] [lecture notes]
- Homework: [sheet5]
Week 6
- Lecture 14 (Mon, Sep 27): Matrix multiplication [recording] [lecture notes]
- Lecture 15 (Wed, Sep 29): Invertibility of matrices and elementary matrices [recording] [lecture notes]
- Lecture 16 (Fri, Oct 1): Invertibility of linear maps and isomorphic vector spaces [recording] [lecture notes]
- Homework: [sheet6]
Week 7
- Lecture 17 (Mon, Oct 4): Invertibility and basis change [recording] [lecture notes]
- Lecture 18 (Wed, Oct 6): Quotient spaces [recording] [lecture notes]
- Lecture 19 (Fri, Oct 8): Rank of a matrix [recording] [lecture notes]
- Homework: [sheet7]
Week 8
- Lecture 20 (Mon, Oct 11): Invariant subspaces, eigenvalues, eigenvectors [recording] [lecture notes]
- Lecture 21 (Wed, Oct 13): Existence of eigenvalues and upper-triangular matrices [recording] [lecture notes]
- Lecture 22 (Fri, Oct 15): Eigenspaces and diagonalization [recording] [lecture notes]
- Homework: [sheet8]
Week 9
- Lecture 23 (Mon, Oct 18): Inner product spaces [recording] [lecture notes]
- Lecture 24 (Wed, Oct 20): Inequalities in inner product spaces [recording] [lecture notes]
- Lecture 25 (Fri, Oct 22): Orthonormal bases [recording] [lecture notes]
Week 10
- Lecture 26 (Mon, Oct 25): Orthogonal complements [recording] [lecture notes]
- Lecture 27 (Fri, Oct 29): Orthonormal projections [recording] [lecture notes]
- Homework: [sheet10]
Week 11
- Lecture 28 (Mon, Nov 1): Adjoint of a map [recording] [lecture notes]
- Lecture 29 (Wed, Nov 3): Self-adjoint and normal operators [recording] [lecture notes]
- Lecture 30 (Fri, Nov 5): Spectral theorems [recording] [lecture notes]
- Homework: [sheet11]
Week 12
- Lecture 31 (Mon, Nov 8): Positive operators and isometries [recording] [lecture notes]
- Lecture 32 (Wed, Nov 10): Polar decomposition and singular value decomposition [recording] [lecture notes]
- Lecture 33 (Fri, Nov 12): Generalized eigenvectors and nilpotent operators [recording] [lecture notes]
Week 13
- Lecture 34 (Mon, Nov 15): Generalized eigenspaces and multiplicities of eigenvalues [recording] [lecture notes]
- Lecture 35 (Fri, Nov 19): Characteristic and minimal polynomial [recording] [lecture notes]
- Homework: [sheet13]
Week 15
- Lecture 36 (Mon, Nov 29): Jordan normal form [recording] [lecture notes]
- Lecture 37 (Wed, Dec 1): Trace of matrices and operators [recording] [lecture notes]
- Lecture 38 (Fri, Dec 3): Determinant of matrices and operators [recording] [lecture notes]
- Homework: [sheet15]
Week 16
- Lecture 39 (Mon, Dec 6): Existence and uniqueness of the determinant [recording] [lecture notes]
- Lecture 40 (Wed, Dec 8): Determinant formulas [recording] [lecture notes]
Literature
The main textbooks for this course are:
- Axler: Linear Algebra Done Right (third edition), Springer, 2015.
The book is available online through SpringerLink. Members of the University of Illinois can also access it online through the University Library (the second edition of the book is also available in hard-copy). - Beezer: A First Course in Linear Algebra, free online book (GNU Free Documentation License Ver 1.2), 2004-2015.
Other good resources for studying linear algebra:
- Meckes and Meckes: Linear Algebra, Cambridge University Press, 2018.
- Friedberg, Insel, Spence: Linear Algebra (fifth edition), Pearson, 2019.
Contact
Email: <mylastname>@illinois.edu
Homepage: felixleditzky.info
Postal address:
Illini Hall, Office 341B
725 S. Wright Street
Champaign, IL 61820
USA