Math 2R03 - Theory of Linear Algebra
(Fall 2024)
MATH 2R03 is a second course in linear algebra. From the calendar:
- Abstract vector spaces. Linear transformations. Inner product spaces. Spectral theorems. Orthogonal bases, other topics. Three lectures, one tutorial; one term
- to learn topics in linear algebra such as linear transformations, inner product spaces, invariant subspaces, and the spectral theorems.
- to learn and improve your proof writing skills.
| |||||||||||
Course Information
Important Note:
The purpose of this webpage is to collect together all of the needed resources for the course. You should also consult the Avenue-To-Learn page and Mosaic where you will find room information.
Instructor:
Office: Hamilton Hall 419 NOTE: I'm currently using HH 326
Office Hours: Wednesdays 9:30-10:20
Email: vantuyl@math.mcmaster.ca
TA:
Silas Vriend
Email: vriendsp@mcmaster.ca
Office Hours: Thursdays 12:30-1:30; Fridays 12:00-1:00 in Math Cafe
Course Outline:
The official course outline is found on SimpleSyllabus
Place and Time:
This course will be offered in-person. The times are:
Lecture C01: Monday, Wednesday 8:30-9:20, Friday 10:30-11:20
Tutorial T01 : Monday 10:30-11:20
Tutorial T02 : Tuesday 1:30-2:20
For locations of the class and tutorials, see Mosaic
Textbook:
Linear Algebra Done Right (Fourth Edition)
by Sheldon Axler
This book can be download for free.
Here is a link to the book on the campus bookstore website: Purchase at Campus Bookstore
Important Links:
The following online resources will be used during this course:
- Crowdmark: We will be using Crowdmark for homework.
- Math 2R03 YouTube Channel: This YouTube channel contains the videos for the lectures. Note that the videos were for 2022 -- my lectures in 2024 will be similar but not an exact copy.
- Avenue to Learn: This is McMaster's learning management system; while we may use it, we will primarily use this webpage.
News (Last Updated: Dec. 17, 2024)
- December 17, 2024 The final exam was last Friday (December 13). The exams have been graded and the final marks have been submitted. You should be able to see your grade on Mosaic either later today or tomorrow. I hope you enjoyed the course!
- December 2, 2024 We looked at an application of the material of this course by looking at Section 5.D on Gershgorin Disks and how to use them to approximate eigenvalues.
- November 29, 2024 In today's class we discussed the two spectral theorems (see Section 7.B). Both of these results are very important -- they tell us that any symmetric matrix can be diagonalized.
- November 27, 2024 We introduced two special types of operators: the self-adjoint operator and the normal operator.
- November 25, 2024 We started Chapter 7 by going over the definition of an adjoint operator.
- November 22, 2024 Midterm 2 marks were released. We also finished Chapter 6 by going over the minimizatin theorem and projection maps
- November 20, 2024 We learned about one of my favourite theorems in the course: the Reisz Representation Theorem (see Section 6.B). I have also uploaded a review sheet for the final. A pratice final is on the A2L page.
- November 18, 2024 Today we learned about the Gram-Schmidt process for any inner product space.
- November 15, 2024 Midterm 2 was today.
- November 13, 2024 We continued going over the basics of inner product spaces (Section 6.A).
- November 11, 2024 We started our discussion of inner product spaces (see Chapter 6).
- November 8, 2024 We finished our discussion of generalized eigenspaces (Section 8.B).
- November 7, 2024 The remaining homework assignments have been added to the web.
- November 6, 2024 We discussed one of the main results of the course, namely, every operator over a complex vector space allows us to decompose our vector space into generalized eigenspaces (see Section 8.B).
- November 4, 2024 We finished our discussion of generalized eigenvectors.
- November 1, 2024 We jumped to Chapter 8 to look at generalized eigenvectors.
- October 31, 2024 Happy Halloween! Solutions to homework 5 have been posted to A2L. Midterm 2 covers up to the material of October 30.
- October 30, 2024 We finished Section 5.D on the diagonalizable operators.
- October 28, 2024 We looked at Section 5.C on operators and upper triangular matrices.
- October 25, 2024 We finished up our discussion minimal polynomials. I also posted a review sheet for Midterm 2 (which is in three weeks).
- October 23, 2024 We introduced minimal polynomials (see Section 5.B).
- October 22, 2024 On Monday we continued our discussion of eigenvalues eigenvectors.
- October 15, 2024 Happy reading week -- just tidying up this webpage.
- October 11, 2024 Today I introduced eigenvalues and eigenvectors of a linear transformation (Section 5.A). Just a reminder that I extended the deadline for the next homework assignment. Have a great fall break!
- October 9, 2024 We did a quick review of polynomials (Chapter 4). Midterm 1 was also returned -- see the solutions on A2L.
- October 7, 2024 We finished our discussion of Chapter 3.D on isomorphisms, and how to view a linear map as a matrix multiplication.
- October 4, 2024 Midterm 1 was today.
- October 2, 2024 We went over the material of Section 3.D on Invertible Linear Maps.
- September 30, 2024 No class today; make sure to study for midterm 1 on Friday.
- September 27, 2024 We looked at Section 3.B (some consequences of the Fundamental Theorem of Linear Maps) and 3.C (associating a matrix to a linear map).
- September 26, 2024 I added information about the midterm (along with a practice test) to A2L. Note that the midterm will not be in our classroom, but in a different room. See your room assignment on A2L.
- September 25, 2024 We covered Section 3.B on the null space and range of a linear map. We also proved the Fundamental Theorem of Linear Maps.
- September 23, 2024 We started Chapter 3 with a discussion of Section 3.A.
- September 20, 2024 We covered Section 2.C in class; the first midterm will cover up to the material in today's class.
- September 19, 2024 We learned about bases of vector spaces (Secton 2.B).
- September 17, 2024 Homework 4 has been added to this website if you want to get ahead.
- September 16, 2024 We finished Section 2.A with an introduction of linearly independent vectors.
- September 13, 2024 We covered the first half of Section 2.A where I introduced the span of a set of vectors.
- September 12, 2024 The TA's office hours have now been posted (see above).
- September 11, 2024 We finished Section 1.C, and I gave some pointers on writing proofs. Make sure you read my Guide to Writing Proofs. I have also found a note taker -- the notes will appear on A2L in the next couple of days, along with my extra notes from today's class.
- September 9, 2024 We started Section 1.C.
- September 6, 2024 We went over the material of Section 1.B. Make sure you try the pratice assignment. I have also finalized my office hours (Wednesdays 9:30-10:20; right after class). The midterms will be on Friday, October 4 (10:30-11:20) and Friday, November 15 (10:30-11:20). Loction will be confirmed later.
- September 4, 2024 Today was our first lecture. I went over the course outline, and the material of Section 1.A.
- August 28, 2024 Webpage started
Return to TOP
Schedule
We will be using the following schedule.
Note 1: There may be changes so please refer back to this page to stay up-to-date
Note 2: The videos are provided as a resource -- they are based upon the third edition of the textbook (we are using the fourth edition). Our lectures may cover different topics. A detailed schedule is given below.
| Math 2R03 Schedule | |||
|---|---|---|---|
| Week 1 (Sept 2-6) | |||
| Homework: Assignment 0 (Due Sept 6) | |||
| Lecture | Topic | Video | Suggested Problems |
| Lecture 1 | 1.A R^n and and C^n | Video | 1.A - 10, 11, 15 |
| Lecture 2 | 1.B Vector Spaces | Video | 1.B - 1, 2, 3 |
| Week 2 (Sep 9-13) | |||
| Homework: Assignment 1 (Due Sep 13) | |||
| Lecture 3 | 1.C Subspaces I | Video | 1.C - 6, 10, 23 |
| Lecture 4 | 1.C Subspaces II Writing Proofs |
Video | 1.C - 10, 19, 20 |
| Lecture 5 | 2.A Span and Linear Independence I |
Video | 2.A - 2, 3, 7 |
| Week 3 (Sep 16-20) | |||
| Homework: Assignment 2 (Due Sep 20) | |||
| Lecture 6 | 2.A Span and Linear Independence II | Video | 2.A - 3, 13 |
| Lecture 7 | 2.B Bases | Video | 2.B - 3, 7, 10 |
| Lecture 8 | 2.C Dimension | Video | 2.C - 11, 14, 17 |
| Week 4 (Sep 23-27) | |||
| Homework: Assignment 3 (Due Sep 27) | |||
| Lecture 9 | 3.A Vector Space of Linear Maps | Video | 3.A - 2, 4, 7 |
| Lecture 10 | 3.B Null Spaces and Ranges I | Video | 3.B - 2, 6, 9 |
| Lecture 11 | 3.B Null Spaces and Ranges II 3.C Matrices |
Video | 3.C -2, 5, 6 |
| Week 5 (Sep 30-Oct 4) | |||
| Homework: Study for Midterm 1 (Oct 4) | |||
| No class Sep 30 | |||
| Lecture 12 | 3.D Invertibility and Isomorphic Vector
Spaces I |
Video | 3.D - 2, 3, 5 |
| Lecture 13 | Midterm 1 (Oc. 4) | ||
| Week 6 (Oc 7-11) | |||
| Homework: Assignment 4 (Due Oct 15) | |||
| Lecture 14 | 3.D Invertibility and Isomorphic Vector Spaces II |
Video | 3.D - 11, 14, 15 (Hint: why is T invertible?) |
| Lecture 15 | 4 Polynomials | Video | 4 - 4, 5, 6 |
| Lecture 16 | 5.A Invariant Subspaces I | Video | 5.A - 7, 8, 9 |
| Week 7 (Oct 14-18) | |||
| Reading Week (No Classes) | |||
| Week 8 (Oct 21-25) | |||
| Homework: Assignment 5 (Due Oct 25) | |||
| Lecture 17 | 5.A Polynomials Applied to Operators 5.B Minimal Polynomials |
Video | 5.A - 2, 5, 21, 23 |
| Lecture 18 | 5.B Minimal Polynomials | No Video | 5.B - 1, 3, 4 |
| Lecture 19 |
5.B Minimal Polynomials II | No Video | 5.B - 6, 10, 11 |
| Week 9 (Oct 28-Nov 1) | |||
| Homework: Assignment 6 (Due Nov 1) | |||
| Lecture 20 | 5.C Upper Triangular Matrices | Video Based on 5.B in 3rd ed |
5.C - 1, 3, 6, 8 |
| Lecture 21 | 5.D Diagonalizable Opeators | Video Based on 5.C in 3rd ed |
5.D - 2, 3, 5, 7 |
| Lecture 22 | 8.A Generalized Eigenvectors and Nilpotent Operators I |
Video | 8.A - 4, 6, 7, 8 |
| Week 10 (Nov 4 -Nov 8) | |||
| Homework: Assignment 7 (Due Nov 8) | |||
| Lecture 23 | 8.A Generalized Eigenvectors and Nilpotent Operators II |
Video | 8.A - 12, 13, 17, 21 |
| Lecture 24 | 8.B Generalized Eigenspace Decomposition | Video | 8.B - 2, 4, 5 |
| Lecture 25 | 8.B Generalized Eigenspace Decomposition II | Video | 8.B - 7, 9, 11, 12 | Other | Videos based upon third edition may also be useful, including material on the minimal and characteristic polynomial |
Video Video Video |
| Week 11 (Nov 11-15) | |||
| Homework: Study for Midterm 2 (Nov 15) | |||
| Lecture 26 | 6.A Inner Products and Norms I | Video | 6.A - 3, 4, 5, 9 |
| Lecture 27 | 6.A Inner Products and Norms II | Video | 6.A - 13, 17, 21, 26 |
| Lecture 28 | Midterm II (Nov 15) | ||
| Week 12 (Nov 18-22) | |||
| Homework: Assignment 8 (Nov 22) | |||
| Lecture 29 |
6.B Orthonormal Bases | Video | 6.B - 2a, 3, 7, 8a |
| Lecture 30 | 6.B Linear Functionals | Video | 6.B - 11, 12 |
| Lecture 31 | 6.C Orthogonal Complements and Minimization |
No Video | 6.C - 1, 2, 3, 5 |
| Week 13 (Nov 25-29) | |||
| Homework: nothing due this week, next assingmet due next Monday | |||
| Lecture 32 | 7.A Self-Adjoint and Normal Operators I | Video | 7.A - 1, 2, 3 |
| Lecture 33 | 7.A Self-Adjoint and Normal Operators II | Video | 7.A - 6, 7, 15 |
| Lecture 34 | 7.B The Spectral Theorem | Video | 7.B - 1, 2, 10, 14 |
| Week 14 (Dec 2-4) | |||
| Homework: Assignment 9 (Due Dec 2) | |||
| Lecture 35 | 5.D - Gershgorin Disk Theorem | No video | |
| Lecture 36 | Buffer/Review | No video | |
Return to TOP
Homework
There will be nine homework assignments (your lowest two marks will be dropped). Assignments are posted below. Assignments will be submitted via Crowdmark. You will receive an email to your McMaster account that you will use to upload your assignment. All assignments due by 11:59PM on the due date.
- Assignment 0 (Due: Sep 6)
- Assignment 1 (Due: Sep 13)
- Assignment 2 (Due: Sep 20)
- Assignment 3 (Due: Sep 27)
- Assignment 4 (Due: Oct 15) (Note change of date)
- Assignment 5 (Due: Oct 25)
- Assignment 6 (Due: Nov 1)
- Assignment 7 (Due: Nov 8)
- Assignment 8 (Due: Nov 22)
- Assignment 9 (Due: Dec 2)
-
Guide to writing proofs
A handout on writing proofs I wrote for algebra in 2010. -
Some remarks on writing mathematical proofs
A nice short guide to writing mathematics by John M. Lee.
Return to TOP
Handouts
All class handouts are available as PDF files (except where indicated)
Course Information
Course handout describing the course (available online)
Midterm 1 Review Sheet
Handout describing first midterm.
Midterm 2 Review Sheet
Handout describing second midterm.
Exam Review Sheet
Handout describing final exam.
Return to TOP
Grading Scheme
I will calculate your mark using two different weightings. Your final mark will be the higher of the two weights.
Weighting 1
20% = 9 Assignments (best 7 used)
40% = Midterms (2 x 20%)
40% = Final Exam
Weighting 2
20% = 9 Assignments (best 7 used)
20% = Maximum of Midterm 1 and 2
60% = Final Exam
Return to TOP
McMaster Polices
There are number of important McMaster Policies that you should be aware of. For a complete and up-to-date list of polices, see the official Course Outline.
With respect to MSAFs, missed work will be moved to final, as described in the course outline.
Return to TOP