Math 2R03 - Theory of Linear Algebra
(Winter 2022)
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.
Comparison to Math 2LA3:
This course's primary concern will be the theory of linear algebra. Math 2LA3 focuses more on the applications of linear algebra, and is offered in both terms. At least one (you can take both) of Math 2LA3 or Math 2R03 is required for all Honours Math and Stats programmes. Math 2R03 is required for Honours Math and Stats with a Mathematics subplan. Either linear algebra course is an allowable prerequisite for most third year Math and Stats courses. For Math 3A03, Math 3B03, Math 3GR3, Math 3F03, Math 3FF3 and Math 3QC3, the requirement is Math 2R03 or a grade of at least a B in 2LA3 (starting in 2022-23).
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Course Information
Important Note:
The purpose of this webpage is to collect together all of the needed resources for the course.
Instructor:
Office: Hamilton Hall 419
Office Hours: Fridays 1-2PM
Email: vantuyl@math.mcmaster.ca
TA:
Jananan (Jan) Arulseelan
Email:arulseej@mcmaster.ca
Place and Time:
This course will be offered in-person (online for first four weeks)
Class C01: Tuesday, Thursday, Friday 2:30-3:20 Burke Science Building (BSB) B135
Tutorial T01: Friday 3:30-4:20 ABB 271
Tutorial T02: Wednesday 3:30-4:20 PGCLL M12
Textbook:
Linear Algebra Done Right
by Sheldon Axler
This book can be download for free from the McMaster Library.
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.
- Avenue to Learn: This is McMaster's learning management system; while we may use it, we will primarily use this webpage.
News (Last Updated: January 7, 2022)
- January 7, 2022 Updated website in response to moving online
- December 10, 2021 Website went live
- November 8, 2021 Webpage started
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Schedule
We will be using the following schedule. Please note that there may be changes; please refer back to this page often to stay up-to-date.
Course Delivery:
The course and its tutorials will be delivered online until February 7, and then in-person. Until February 7, the online portion of the course will be delivered using both asynchronous and synchronous components. The asynchronous component consist of video lectures of the course material (posted on Avenue and YouTube). For the synchronous component, we will use the scheduled class time as follows:
- First lecture of the week. (Tuesday) The schedule class time will be used to review the previous week's material, preview the current week's material, and to answer any questions. A link is provided in A2L.
- Second lecture of the week. (Thursday) The time will be used to watch online lectures.
- Third lecture of the week. (Friday) Instructor will be available for office hours. A link is provided in A2L.
A detailed schedule is given below.
| Math 2R03 Schedule | ||
|---|---|---|
| Week 1 (Jan 10-14) | ||
| Homework: Assignment 0 (Due Jan 15) | ||
| Lecture | Topic | Video |
| Lecture 1 | 1.A R^n and and C^n | Video |
| Lecture 2 | 1.B Vector Spaces | Video |
| Lecture 3 | 1.C Subspaces I | Video |
| Week 2 (Jan 17-21) | ||
| Homework: Assignment 1 (Due Jan 22) | ||
| Lecture 4 | 1.C Subspaces II Writing Proofs |
Video |
| Lecture 5 | 2.A Span and Linear Independence I |
Video |
| Lecture 6 | 2.A Span and Linear Independence II | Video |
| Week 3 (Jan 24-28) | ||
| Homework: Assignment 2 (Due Jan 29) | ||
| Lecture 7 | 2.B Bases | Video |
| Lecture 8 | 2.C Dimension | Video |
| Lecture 9 | 3.A Vector Space of Linear Maps | Video |
| Week 4 (Jan 31-Feb 4) | ||
| Homework: Assignment 3 (Due Feb 5) | ||
| Lecture 10 | 3.B Null Spaces and Ranges I | Video |
| Lecture 11 | 3.B Null Spaces and Ranges II 3.C Matrices |
Video |
| Lecture 12 | 3.D Invertibility and Isomorphic Vector Spaces I |
Video |
| Week 5 (Feb 7-11) | ||
| Homework: Study for Midterm 1 (Feb 11) | ||
| Lecture 13 | 3.D Invertibility and Isomorphic Vector Spaces II |
Video |
| Lecture 14 | 4 Polynomials | Video |
| Lecture 15 | Midterm 1 (Feb. 11) | |
| Week 6 (Feb 14-18) | ||
| Homework: Assignment 4 (Due Feb 26) Note Date | ||
| Lecture 16 | 5.A Invariant Subspaces I | Video |
| Lecture 17 | 5.A Invariant Subspaces II 5.B Eigenvectors and Upper-Triangular Matrices I |
Video |
| Lecture 18 |
5.B Eigenvectors and Upper-Triangular Matrics II |
Video |
| Week 7 (Feb 21 - 25) | ||
| Reading Week (No Classes) | ||
| Week 8 (Feb 28-March 4) | ||
| Homework: Assignment 5 (Due March 5) | ||
| Lecture 19 | 5.C Eigenspaces and Diagonal Matrices | Video |
| Lecture 20 | 8.A Generalized Eigenvectors and Nilpotent Operators I |
Video |
| Lecture 21 | 8.A Generalized Eigenvectors and Nilpotent Operators II |
Video |
| Week 9 (March 7-11) | ||
| Homework: Assignment 6 (Due March 12) | ||
| Lecture 22 | 8.B Decomposition of an Operator I | Video |
| Lecture 23 | 8.B Decomposition of an Operator II | Video |
| Lecture 24 |
8.C Characteristic and Minimal Polynomials |
Video |
| Week 10 (March 14-18) | ||
| Homework: Study for Midterm 2 (March 18) | ||
| Lecture 25 | 8.D Jordan Form | Video Extra Video |
| Lecture 26 | 6.A Inner Products and Norms I | Video |
| Lecture 27 | Midterm II (March 18) | |
| Week 11 (March 21-25) | ||
| Homework: Assignment 7 (March 26) | ||
| Lecture 28 | 6.A Inner Products and Norms II | Video |
| Lecture 29 |
6.B Orthonormal Bases | Video |
| Lecture 30 | 6.B Linear Functionals | Video |
| Week 12 (March 28-April 1) | ||
| Homework: Assignment 8 (April 2) | ||
| Lecture 31 | 7.A Self-Adjoint and Normal Operators I | Video |
| Lecture 32 | 7.A Self-Adjoint and Normal Operators II | Video |
| Lecture 33 | 7.B The Spectral Theorem | Video |
| Week 13 (April 4-8) | ||
| Homework: (Nothing due this week) | ||
| Lecture 34 | 7.C Positive Operators and Isometries | Video |
| Lecture 35 | 7.D Polar Decompositions | Video |
| Lecture 36 | 7.D Singular Value Decomposition | Video |
| Week 14 (April 11-12) | ||
| Homework: Assignment 9 (Due April 12) | ||
| Lecture 37 | Buffer/Review | |
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Homework
There will be nine homework assignments (your lowest mark 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: Jan 15)
- Assignment 1 (Due: Jan 22)
- Assignment 2 (Due: Jan 29)
- Assignment 3 (Due: Feb 5)
- Assignment 4 (Due: Feb 26)
- Assignment 5(Due: March 5)
- Assignment 6(Due: March 12)
- Assignment 7 (Due: March 26)
- Assignment 8 (Due: April 2)
- Assignment 9 (Due: April 12) (Note date)
-
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.
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Handouts
All class handouts are available as PDF files (except where indicated)
Course Information
Course handout from first day of class
Midterm 1 Review Sheet
Handout describing first midterm.
Midterm 2 Review Sheet
Handout describing second midterm.
Exam Review Sheet
Handout describing final exam.
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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 8 used)
40% = Midterms (2 x 20%)
40% = Final Exam
Weighting 2
20% = 9 Assignments (best 8 used)
20% = Maximum of Midterm 1 and 2
60% = Final Exam
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McMaster Polices
1. McMaster University Statement on Inclusivity and Academic Integrity. The University values integrity, inclusiveness and teamwork, and strives to support the personal and collective growth of the McMaster student community.These values are foundational to ensuring campus environments -- both in-person and virtual -- are conducive to personal wellbeing and academic success.
2. Inclusivity and a Culture of Respect. As a McMaster student, you have the right to experience and the responsibility to demonstrate respectful and dignified interactions within all of our living, learning and working communities. Expectations are described in Code of Student Rights and Responsibilities
It is essential that students be mindful of their interactions online, as the Code remains in effect in virtual learning environments. The Code applies to any interactions that adversely affect, disrupt, or interfere with reasonable participation in University activities. Student disruptions or behaviours that interfere with university functions on online platforms (e.g. use of Avenue 2 Learn, WebEx or Zoom for delivery), will be taken very seriously and will be investigated. Outcomes may include restriction or removal of the involved students' access to these platforms. Additional information about the Code and netiquette can be found here.
3. Academic Integrity and Honesty. As a McMaster student, you are expected to exhibit honesty and ethical behaviour in all aspects of the learning process. The academic credentials that you earn are rooted in the principles of honesty and academic integrity.
Academic dishonesty is to knowingly act or fail to act in a way that results or could result in unearned academic credit or advantage. This behaviour can result in serious consequences, e.g. the grade of zero on an assignment, loss of credit with a notation on the transcript (notation reads: Grade of F assigned for academic dishonesty), and/or suspension or expulsion from the university.
It is your responsibility to understand what constitutes academic
dishonesty. For information on the various types of academic
dishonesty
please refer to the Academic Integrity Policy, located at:
http://www.mcmaster.ca/academicintegrity/
The following illustrates only three forms of academic dishonesty:
- plagiarism, e.g. the submission of work that is not one's own or for which other credit has been obtained.
- improper collaboration in group work,
- copying or using unauthorized aids in tests and examinations.
4. Academic Accommodation of Students with Disabilities. Students with disabilities who require academic accommodation must contact Student Accessibility Services (SAS) to make arrangements with a Program Coordinator. Student Accessibility Services can be contacted by phone 905-525-9140 ext. 28652 or e-mail sas@mcmaster.ca. For further information, consult McMaster University's Academic Accommodation of Students with Disabilities policy.
5. Requests for Relief for Missed Academic Term Work. If you have missed work, it is your responsibility to take action. If you are absent from the university for medical and non-medical (personal) situations, lasting fewer than 3 days, you may report your absence, once per term, without documentation, using the McMaster Student Absence Form (MSAF). See Requests for Relief for Missed Academic Term Work
Absences for a longer duration or for other reasons must be reported to your Faculty/Program office, with documentation, and relief from term work may not necessarily be granted. In Math 2R03, the percentages of the missed work will be transferred to the final examination. Please note that the MSAF may not be used for term work worth 25% or more, nor can it be used for the final examination.
6. Academic Accommodation for Religious, Indigenous or Spiritual Observances (RISO). Students requiring academic accommodation based on religious, indigenous or spiritual observances should follow the procedures set out in the RISO policy. Students requiring a RISO accommodation should submit their request to their Faculty Office normally within 10 working days of the beginning of term in which they anticipate a need for accommodation or to the Registrar's Office prior to their examinations. Students should also contact their instructors as soon as possible to make alternative arrangements for classes, assignments, and tests.
7. Important Message. The instructor and university reserve the right to modify elements of the course during the term. The University reserves the right to change the dates and deadlines for any or all courses in extreme circumstances (e.g., severe weather, labour disruptions, etc.). Changes will be communicated through regular McMaster communication channels, such as McMaster Daily News, A2L and/or McMaster email. If either type of modification becomes necessary, reasonable notice and communication with the students will be given with explanation and the opportunity to comment on changes. It is the responsibility of the student to check their McMaster email and course websites weekly during the term and to note any changes.
8. On-line Statement for Courses Requiring Online Access or Work. In this course we will be using Zoom, Avenue-to-Learn, YouTube, and Crowdmark. Students should be aware that, when they access the electronic components of this course, private information such as first and last names, user names for the McMaster e-mail accounts, and program affiliation may become apparent to all other students in the same course. The available information is dependent on the technology used. Continuation in this course will be deemed consent to this disclosure. If you have any questions or concerns about such disclosure please discuss this with the course instructor.
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