Abstract Linear Algebra, Math 416. Spring 2025. Section X-13.

Main goals and topics:

Systems of linear equations are ubiquitous in all areas of science. To understand functions or systems of equations that are non-linear, sometimes our best hope is to approximate them using linear versions, and analyse the linear version. In this course we will learn how to analyse such linear versions. The topics of this course are basic tools in most areas of mathematics, and also, physics, computer science, engineering, and related areas. Math 416 is a rigorous proof based treatment of linear algebra. The main concepts that we will cover are

• systems of linear equations, row reduction and echelon form

• vectors and matrices, matrix multiplication, invertibility. and inverses

• vectors spaces and linear transformations

• subspaces, linear combinations, spanning sets and bases

• representing linear transformations as matrices, change of basis

• kernel and image, row and column rank, Rank-Nullity theorem

• determinants

• eigenvalues and eigenvectors

• finding eigenvalues of a transformation using the characteristic polynomial, minimal polynomial

• finding eigenspaces associated to eigenvalues

• inner product spaces, their algebra and geometry, Cauchy-Schwarz inequality

• orthogonal projections, Gram-Schmidt, least squares

• orthogonal and unitary matrices, spectral theory

• bilinear forms

• Jordan form   

Important details:

Instructor: Neeladri Maitra

Contact: By email: nmaitra@illinois.edu. Please include [math 416, X-13] in the subject. Make sure to email from your `illinois.edu' account.

Classes: MWF, 12:00 - 12:50, 148 Armory (spare room: 147 Altgeld Hall, which we will use in case of any problem with the Armory hall; in case an upcoming lecture is scheduled not at Armory but at Altgeld Hall, I will notify through canvas).

Canvas: Make sure that you are added to the canvas page of the course. Please write to me if you are not added but want to be. I encourage you to have discussions among each other at canvas.

Office: 

Office hours: 

Grader: 

Resources: Textbook is Linear Algebra by Friedberg, Insel, Spence, 5th edition. Here is an e-copy of the 4th edition.

We will sometimes refer to the freely available online book A First Course in Linear Algebra, by Beezer (version 3.5, 2015). I will also post my handwritten lecture notes after every class (see 'Class schedule' below).

Prerequisite: Math 241 required. Math 347 strongly recommended.

e-devices: Please make sure to turn off your cell-phones during the lecture. Usage of calculator is not allowed for exams.

Personal remarks:

Linear Algebra is an essential tool and a stepping stone towards research level mathematics, physics, statistics, engineering or theoretical computer science courses. If you aspire to continue in any branch of science at a research academic level, consider taking the course seriously, and participate and discuss as much as you can. Getting the fundamentals from this course clear will make life easy in the future. :-) 

Grading, exams, homework policy: 

Homeworks: 

• Homeworks will be posted weekly at Canvas on Fridays 4 PM. These are due on next Thursdays 9 PM, to be handed through Canvas. 

• Submit either pdf files or combined images. If your homework is handwritten, please make sure they are legible. Make sure homework problems are submitted in the right order. Consider using a free mobile scanner app (e.g., camscanner, etc.) to turn images into pdf files. Again, please double check to make sure your solutions are readable.

• If you wish to continue in either math or physics, I encourage you to learn LATEX and typing up your homeworks in LATEX. LATEX is an essential skill, that will be useful later. You can create a free account on  https://overleaf.com/, and here is a really good resource, in particular, check out Chapter 3 on typesetting math. Feel free to search and learn from other resources too.

• I highly encourage you to exchange ideas, collaborate among each other, but please make sure you understand, and write your own solution.

• No late homeworks please. They won't be graded. In case you will be absent when the homework is due, please turn them in advance.

• Your three lowest homeworks will be dropped.

Exams:

• Midterms: There will be three in-class midterms. 

• Final exam: TBA.

No make-up exams: There will not be any make-up exams after the actual exams. However, if there is a conflict with a university related activity like a competition or a conference, a make-up exam in advance may be arranged. In case of any documented emergency or illness, an exam can be dropped.

Grading: 

Class schedule (tentative):

1. Wednesday, January 22, Armory 148. Introduction. [lecture notes]

2. Friday, January 24, Altgeld Hall 147. Vector spaces. [lecture notes]

3. Monday, January 27, Armory 148. Subspaces. [lecture notes]

4. Wednesday, January 29, Armory 148. Linear combinations. [lecture notes]

5. Friday, January 31, Armory 148. Linear equations. [lecture notes]

6. Monday, February 3, Armory 148. Gauss-Jordan elimination. [lecture notes]

7. Wednesday, February 5, Armory 148. Solution spaces to linear systems. [lecture notes]

8. Friday, February 7, Armory 148. Linear dependence and independence. [lecture notes]

9. Monday, February 10, Armory 148. Basis and dimension 1. [lecture notes]

10. Wednesday, February 12, Armory 148. Basis and dimension 2. [lecture notes]

11. Friday, February 14, Armory 148. Basis and dimension 3. [lecture notes]

12. Monday, February 17, Armory 148. Review and problems.

13. Wednesday, February 19, Armory 148. Mid Term 1.

14. Friday, February 21, Armory 148. Linear Transformations. [lecture notes]

15. Monday, February 24, Armory 148. Rank Nullity Theorem. [lecture notes]

16. Wednesday, February 26, Armory 148. Encoding linear-transformations as matrices. [lecture notes]

17. Friday, February 28, Armory 148. Compositions of linear transformations. [lecture notes]

18. Monday, March 3, Armory 148. Matrix multiplication. [lecture notes]

19. Wednesday, March 5, Armory 148. Invertibility and isomorphisms. [lecture notes]

20. Friday, March 7, Armory 148. Matrix inverses and rank. [lecture notes]

21. Monday, March 10, Armory 148. Coordinates. [lecture notes]

22. Wednesday, March 12, Armory 148. Mid Term 2.

23. Friday, March 14, Armory 148. Introduction to determinants. [lecture notes]

24. Monday, March 17, no class (spring break).

25. Wednesday, March 19, no class (spring break).

26. Friday, March 21, no class (spring break).

27. Monday, March 24, Armory 148. Definition of a determinant. [lecture notes]

28. Wednesday, March 26, Armory 148. Properties of the determinant. [lecture notes]

29. Friday, March 28, Armory 148. Elementary matrices, volumes. [lecture notes]

30. Monday, March 31, Armory 148. Diagonalization and Eigenstuff. [lecture notes]

31. Wednesday, April 2, Armory 148. Finding eigenvectors. [lecture notes]

32. Friday, April 4, Armory, 148. Diagonalizability criteria. [lecture notes]

33. Monday, April 7, Armory, 148. Proof of diagonalizability criterion. [lecture notes]

34. Wednesday, April 9, Armory 148. Mid Term 3.

35. Friday, April 11, Armory 148. Markov Chains. [lecture notes] 

36. Monday, April 14, Armory 148. More on Markov Chains. [lecture notes]

37. Wednesday, April 16, Armory 148. Inner product spaces. [lecture notes]

38. Friday, April 17, Armory 148. More on inner products. [lecture notes]

39. Monday, April 21, Armory 148. Gram-Schmidt orthonormalization. [lecture notes]

40. Wednesday, April 23, Armory 148. Orthogonal projections. [lecture notes]

41. Friday, April, 25, Armory 148. Adjoints. [lecture notes]

42. Monday, April 28, Armory 148. Normal and self-adjoint operators. [lecture notes]

43. Wednesday, April 30, Armory 148. Diagonalizing self-adjoint operators. [lecture notes]

44. Friday, May 2, Armory 148. Orthogonal and unitary matrices. [lecture notes]

45. Monday, May 5, Armory 148. Jordan form, SVD. [lecture notes]

46. Wednesday, May 7, Armory 148. Cayley-Hamilton theorem. [lecture notes]

47. Friday, May 9, Armory 148. Final - 13:30 - 16:30.  

Academic integrity: 

Violations of academic integrity will be taken seriously, and will be handled under Article I, part 4 of the student code.

Resources for students with disabilities: 

To ensure that disability related concerns are properly addressed from the beginning, students with disabilities who require reasonable accommodation should contact Disability Resources and Educational Services (DRES) and me as soon as possible. To contact DRES, visit 1207 S. Oak St., Champaign, call 333-4603, email disability@illinois.edu, or visit https://dres.illinois.edu/.

Resources and supporting fellow students in distress:

If you come across a classmate whose behaviour concerns you, whether in regards to their well-being or yours, we encourage you to refer this behaviour to the Student Assistance Center (333-0050), or online at https://odos.illinois.edu/community-of-care/referral. Staffs in the Student Assistance Center will then reach out to students to make sure they have the support they need to be healthy and safe.

Struggles with mental health can have an impact on your experience at Illinois; significant stress, strained relationships, anxiety, excessive worry, alcohol/drug problems, a loss of motivation, or problems with eating and/or sleeping can all interfere with optimal academic performance. All students are encouraged to reach out to talk with someone. Support is available at the Counseling Center (https://counselingcenter.illinois.edu/) or McKinley Health Center (https://mckinley.illinois.edu/). For mental health emergencies, you can 911, or walk-in to the Counseling Center, no appointment required.