Winter 2022: MAT A22 Linear Algebra I for Mathematical Sciences
From the Course Calendar:
“A conceptual and rigorous approach to introductory linear algebra that focuses on mathematical proofs, the logical development of fundamental structures, and essential computational techniques. This course covers complex numbers, vectors in Euclidean $n$space, systems of linear equations, matrices and matrix algebra, Gaussian reduction, structure theorems for solutions of linear systems, dependence and independence, rank equation, linear transformations of Euclidean $n$space, determinants, Cramer’s rule, eigenvalues and eigenvectors, characteristic polynomial, and diagonalization.”
This course leads up to MAT B24 Linear Algebra II:
“Fields, vector spaces over a field, linear transformations; inner product spaces, coordinatization and change of basis; diagonalizability, orthogonal transformations, invariant subspaces, CayleyHamilton theorem; hermitian inner product, normal, selfadjoint and unitary operations. Some applications such as the method of least squares and introduction to coding theory.”
Course Coordinator’s Message
I’m glad that you’re taking this course. It is a really special course, because linear algebra powers everything from Google’s PageRank algorithm to the shading of pixels in video games. Moreover, this course in linear algebra introduces you to mathematics proofs in context. Linear algebra has some of the nicest or cleanest proofs in mathematics. It is the model of a nice simple theory with surprisingly powerful applications.
This course leads to MAT B24 and also deep ideas from MAT C01. It is the beginning of your journey in to algebra. I love linear algebra because it is magic that works. I hope that in this course you’ll experience some of the joy and wonder that this subject contains.
Policy on InPerson vs Online Delivery
We are making every effort to provide an inperson educational experience. The Ontario government has asked us to remain online until January 31st. That means, we’ll be online for at least the first three weeks of classes. We are planning to have inperson midterms and an inperson exam. However, we recognize the threat posed by omicron variant of COVID19. As we get more information from the Registrar and the Ontario government, we will keep you updated and informed.
Currently, we plan for:
Activity  Mode of Delivery 

Lecture  Online until January 31st, and inperson afterwards 
Tutorial  Online until January 31st, and inperson afterwards 
Term Tests  Inperson with dates to be announced 
Exam  Inperson with date to be announced 
Frequently Asked Questions
 When do tutorials start? Tutorials start in Week 2.
 Will the course be online or inperson? We will try to make as much of this course inperson.
 Will the online components be recorded? Lectures will be recorded and posted on the Media Gallery, but tutorials will not be recorded.
 Do I have to attend the lecture synchronously? No, you do not need to attend synchronously as there is no lecture participation component. However, we encourage
 Do I have to attend tutorials synchronously? No, you do not. However, they will not be recorded and you will not be able to ask questions of the TAs.
Course Staff
Proofessors

Parker GlynnAdey (Course Coordinator)
 Call Me: “Parker” or “Proofessor Parker”
 EMail : parker.glynn.adey@utoronto.ca (See Communication Policy.)
 Website: http://pgadey.ca/
 Office: IC 344

Kaidi Ye
 EMail: kaidi dot ye at mail dot utoronto dot ca
Communication Policy
Piazza
This term we will be using Piazza for class discussion. The system is highly catered to getting you help fast and efficiently from classmates, the TAs, and myself. Rather than emailing questions to the teaching staff, I encourage you to post your questions on Piazza. If you have any problems or feedback for the developers, email team@piazza.com.
Find our class signup link at: https://piazza.com/utoronto.ca/winter2022/mata22h3s20221
Please include your name and student number in every email that you send. Mail must be from an official University of Toronto account. To make sure that your email does not get lost you must include this magic formula in the body:
Be sure to include the precise question, and the problem or difficulty. If you’re not able to write out the question, take a photo or attach a PDF.
Above all, don’t worry about emailing me or any of the course staff. We are not evil trolls. We won’t get angry if you email us. Answering student emails is a part of our job.
However, email is only part of our job. We might not respond to your email on the same day that you send it. Generally, give us at least two business days to respond. Parker has limited access to his computer on Tuesdays, Thursdays, and weekends.
Here is an example of a wellformatted email:
To: parker.glynn.adey@utoronto.ca
From: leonhard.euler@utoronto.ca
Subject: [MAT A22] What is a vector?
Hi! I am Leonhard Euler (12932188) from MAT A22.
I need help with this question: Find a vector orthogonal to (1,1).
My problem is this: I don’t know what the word "vector" means.
Thanks!
A22Winter2022:8620406
All emails must include the following:
 University of Toronto Email account
 Name and student number
 The magic formula: A22Winter2022:8620406
Textbooks
We are going to use two books for this course, both of them are free online.
Primary book: Linear Algebra with Applications by Keith Nicholson. This is a book with lots of computations and examples. It is suitable for MAT A23 or MAT A22. We will use it as the primary reference for the course, and follow its notation and section order.
Secondary book: Linear Algebra Done Right by Sheldon Axler. This is a formal and proof oriented introduction to linear algebra. We will use it as a reference, and will sometimes use it as a source of homework problems.
Grading Scheme
Task  Weight 

Exam  40% 
Term Tests  2x20% 
Assignments  (61)x4% 
Assignments
Goal: these assignments give you the opportunity to deepen your understanding of topics covered in this course, and to practice. We use these assignments to determine if you can solve problems slowly, without time constraints.
Procedure: we will be using Crowdmark to grade assignment submissions. You will get a personalized submission link sent to your UToronto email address. Do NOT share this link with other students.
Submission Guidelines: Assignments need to be submitted online through Crowdmark. You will have a week to write the assignment.
Evaluation Criteria: The TAs will grade two of the five questions. Present your solutions in a logical and clear manner. Detailed solutions will be made available shortly after the deadline of submission.
Please pay attention to the following when writing assignments:
 Format: solutions are neatly and correctly assembled with professional look. The graders should not struggle to read your work.
 Completeness: all steps are clearly and accurately explained.
 Content: the written solutions demonstrate mastery and fluency with the content of the course.
Common Questions about Assignments

Why does it say (61)x4% for assignments? This is a short way of writing that there are six assignments, but one of them will be dropped. It’s a saying that there are 61 = 5 assignments that count for 4% each.

Which assignment will get dropped? Your lowest assignment will be dropped automatically. You do not need to request this, or send an email about it.

What happens if I miss an assignment deadline? You will not be penalized for missing an assignment deadline, if you submit before the solutions are released. We understand that uploading to Crowdmark is difficult and there are technical mistakes. However, if you submit you assignment after the solutions are released then you will receive zero on the assignment.
Term Tests
Goal: these written tests give you the opportunity to demonstrate your understanding of core concepts and topics in a written format. You will gain experience of communicating mathematical ideas in a logical manner. We write tests in a limited amount of time to assess your fluency with the material.
Procedure: We will hold term tests outside of class time. We will post an announcement on Quercus about where and when to write the term tests. They will be written inperson and invigilated. You will have two hours to write each test.
Evaluation Criteria: In general, you need to present your solutions in a logical and clear manner. Detailed solutions will be made available shortly after the tests.
Common Questions about Term Tests
 What happens if the term tests are online? If the term tests are online, they will be written synchronously and invigilated via Zoom. You will need to have reliable internet connection and a private working environment. They will still last two hours, and you will submit your work by Crowdmark.
 When will we know the date and time of the term tests? You will know the date and time of the terms at least one week before the test. We are waiting for information from the Registrar about when and where we can hold term tests.
 Will the tests be hard or easy? This depends on your level of fluency with the material. Some student will find the tests hard, and other students will find the tests easy. We have designed the tests to accurately assess the learning objectives, and believe that they are a fair reflection of the course.
 What should I do to prepare for the test? Practice. Practice. Practice! You can practice via the Suggested Questions from the Reading Guide, ask questioning in lecture, and attending and participating in Tutorials.
 Will all sections of the course write the same test? Yes. All sections of the course will write the same test, so that everyone gets a fair and equal grade. All sections of the course will cover the same material.
Policy on Missed Term Tests
Life is full of unexpected complications. Sometimes, people miss term tests.
If you miss the first term test, then your grading scheme becomes:
Task  Weight 

Exam  40% 
Second Term Test  30% 
Homework  (61)x6% 
If you miss the second term test, then your grading scheme becomes:
Task  Weight 

Exam  40% 
First Term Test  20% 
Homework  (61)x8% 
If you miss both term tests, then you must meet with Parker to discuss alternatives for you.
How Does Grading Work?
The professors set the assignments and term tests. You, the student, do your best work and hand it in. The TAs then grade your term tests and assignments. Please note, that the professors do not directly look at your homework. The TAs then return your graded work, and you may request a regrade or further comments if you think the grading is unclear.
For assignment, the TAs will only grade two of the five questions. This policy of subset grading helps us to save time and energy, and teaches you to evaluate your own work.
For term tests, the TAs will grade all the questions.
Common Questions about Grading
 Why do you only grade two of the five questions? There are three reasons for this. Firstly, we want to save time and energy. Secondly, we think that if you are doing your best work then any two questions should be representative of your whole assignment. Thirdly, we want you to learn to evaluate your own work. We post solutions to all five problems so that you can see how well you did on the three unmarked problems.
 Can you tell me which two questions will be graded? No. We cannot reveal to you which two questions will be graded.
Regrading
You can use the official MAT A22 ReGrade Form.
All requests will be read and considered by Kaidi Ye and Parker GlynnAdey. You may submit a regrade request within oneweek of receiving your grade. We will make sure that you get a response before the final exam. Submit one copy of this form for each regrade request. If you would like three questions regraded, please submit three copies of this form.
Common Questions about Regrading
 How will I know that my assignment was regraded? You will receive an email telling you the outcome of your regrade request.
 Can I ask why I got a paricular grade? Yes, you can use the regrade system to ask for more information about why you got a particular grade. If you want clarification about grading please write: “REQUEST FOR CLARIFICATION”.
Schedule
Themes
 Weeks 1  6: Calculations in Coordinates
 Weeks 7  9: Geometry and Linear Algebra
 Weeks 10  12: Abstract Vector Spaces
WeekByWeek
 Week 1: Complex Numbers and Linear Systems
 Appendix A: Complex Numbers
 1.1 Solutions and Elementary Operations
 Week 2: Gaussian Elimination and Homogeneous Equations
 1.2 Gaussian Elimination
 1.3 Homogeneous Equations
 Week 3: Matrix Algebra
 2.1 Matrix Addition, Scalar Multiplication, and Transposition
 2.2 MatrixVector Multiplication
 2.3 Matrix Multiplication
 Week 4: Matrix Inverses and Linear Transformations
 2.4 Matrix Inverses
 2.5 Elementary Matrices
 2.6 Linear Transformations
 Week 5: Determinants and Eigenvalues
 3.1 The Cofactor Expansion
 3.2 Determinants and Matrix Inverses
 3.3 Diagonalization and Eigenvalues
 Extra: 3.6 Proof of the Cofactor Expansion Theorem
 Week 6: Eigenvalues and Eigenvectors
 Further material on eigenvalues and eigenvectors
 3.4 An Application to Linear Recurrences
 Week 7: Vectors and Lines
 4.1 Vectors and Lines
 4.2 Projections and Planes
 4.3 More on the Cross Product
 Week 8: Subspaces, Independence, and Spans
 5.1 Subspaces and Spanning
 5.2 Independence and Dimension
 5.3 Orthogonality
 Week 9: Similarity and Diagonalization
 5.4 Rank of a Matrix
 5.5 Similarity and Diagonalization
 Week 10: Abstract Vector Spaces
 6.1 Examples and Basic Properties
 6.2 Subspaces and Spanning Sets
 6.3 Linear Independence and Dimension
 Week 11: Linear Transformations Abstractly
 7.1 Examples and Elementary Properties
 7.2 Kernel and Image of a Linear Transformation
 Week 12: The CayleyHamilton Theorem
 Characteristic polynomials
 CayleyHamilton theorem
Suggested Exercises (Up to Week 6)
 Week 1:
 Appendix A: Complex Numbers
 Computations: 2, 3, 4, 12, 19, 20, 22
 Proofs: 14, 15, 16, 17, 21, 25, 29 ($a + b \sqrt{2}$)
 1.1 Solutions and Elementary Operations
 Computations: 1, 7, 8, 9, 10, 15
 Proofs: 6, 12, 13, 14
 Appendix A: Complex Numbers
 Week 2:
 1.2 Gaussian Elimination
 Computations: 2, 3, 4, 5, 16
 Proofs: 8, 9, 12, 14, 22
 1.3 Homogeneous Equations
 Computations: 2, 3, 4, 5
 Proofs: 1, 8, 9, 10, 11, 12
 1.2 Gaussian Elimination
 Week 3:
 2.1 Matrix Addition, Scalar Multiplication, and Transposition
 Computations: 1, 2, 3, 14
 Proofs: 11, 12, 17, 18, 19, 20, 21
 2.2 MatrixVector Multiplication
 Computations: 1, 2, 3, 5, 7, 8, 11, 12*, 13*
 Proofs: 6, 10, 15, 16, 21, 22
 2.3 Matrix Multiplication
 Computations: 1, 4, 5, 8
 Proofs: 6, 7, 9, 14, 15, 17, 23, 24, 27, 30*, 32
 2.1 Matrix Addition, Scalar Multiplication, and Transposition
 Week 4:
 2.4 Matrix Inverses
 Computations: 1, 2, 4, 7, 12 (CayleyHamilton), 14 (group)
 Proofs: 9, 10, 18, 21, 30, 32, 36
 2.5 Elementary Matrices
 Computations: 1, 2, 8
 Proofs: 4, 9, 14, 17 (rowequivalence), 20, 23
 2.6 Linear Transformations
 Computations: 1, 3, 7, 10, 20, 21
 Proofs: 5 (kernel and image), 11, 22
 2.4 Matrix Inverses
 Week 5:
 3.1 The Cofactor Expansion
 Computations: 1, 3, 5, 17
 Proofs: 9, 27 (CayleyHamilton)
 3.2 Determinants and Matrix Inverses
 Computations: 2, 3, 4, 8 (Cramer’s rule)
 Proofs: 10, 14, 16, 20
 3.3 Diagonalization and Eigenvalues
 Computations: 1, 6, 8
 Proofs: 3, 5, 13, 14, 15, 19, 21
 Extra: 3.6 Proof of the Cofactor Expansion Theorem
 3.1 The Cofactor Expansion
 Week 6:
 3.4 An Application to Linear Recurrences
 Computations: 1, 4, 5
 Proofs: 11, 13
 3.4 An Application to Linear Recurrences
 Week 6:
 3.4 An Application to Linear Recurrences
 Computations: 1, 4, 5
 Proofs: 11, 13
 3.4 An Application to Linear Recurrences
 Week 7: Subspaces, Independence, and Spans
 5.1 Subspaces and Spanning
 Computations: 1, 5, 11
 Proofs: 9, 16, 18, 21
 5.2 Independence and Dimension
 Computations: 1
 Proofs: 7, 11, 16, 17
 5.3 Orthogonality
 Computations: 2, 4
 Proofs: 7, 9, 15, 16
 5.1 Subspaces and Spanning
 Week 8: Similarity and Diagonalization
 5.4 Rank of a Matrix
 Computations: 1, 2
 Proofs: 4, 5, 9, 10, 18
 5.5 Similarity and Diagonalization
 Computations: 4
 Proofs: 3, 5, 7, 13
 5.4 Rank of a Matrix
 Week 9: Abstract Vector Spaces
 6.1 Examples and Basic Properties
 Computations: 5, 7
 Proofs: 2, 9, 11
 6.2 Subspaces and Spanning Sets
 Computations: 6, 7, 13
 Proofs: 10, 12, 18, 27
 6.3 Linear Independence and Dimension
 Computations: 5, 8
 Proofs: 13, 14, 32, 35
 6.1 Examples and Basic Properties
 Week 10: Linear Transformations Abstractly
 7.1 Examples and Elementary Properties
 Computations: 4
 Proofs: 1, 2, 14, 22
 7.2 Kernel and Image of a Linear Transformation
 Computations: 1, 14, 20
 Proofs: 5, 7, 29
 7.1 Examples and Elementary Properties
Schedule of Tasks
Week  Task 

1  
2  Assignment 1 
3  
4  Assignment 2 
5  
6  Assignment 3 
Reading Week!  
7  
8  Assignment 4 
9  
10  Assignment 5 
11  
12  Assignment 6 
Exact Dates of Tasks
Task  Date and Time of Task  Date and Time of Solutions 

Assignment 1  Monday January 10th at 13:00 to Thursday January 20th at 13:00  Monday January 24th at 12:45 
Assignment 2  Monday January 24th at 13:00 to Thursday February 3rd at 13:00  Monday February 7th at 12:45 
Assignment 3  Monday February 7th at 13:00 to Thursday February 17th at 13:00  Monday February 21st at 12:45 
Assignment 4  Monday February 28th at 13:00 to Thursday March 10th at 13:00  Monday March 14th at 12:45 
Assignment 5  Monday March 14th at 13:00 to Thursday March 24th at 13:00  Monday March 28th at 12:45 
Assignment 6  Monday March 28th at 13:00 to Thursday April 7th at 13:00  Monday April 11th at 12:45 
Term Test 1  will be announced when the Registrar gives us a date  
Term Test 2  will be announced when the Registrar gives us a date  
Exam  will be announced when the Registrar gives us a date 
To add the dates above to your Google Calendar, import this calendar.
Academic Integrity
The instructional team wants to make sure that everyone has a fair chance to succeed in this course. Therefore, we define an academic integrity violation to be accessing or communicating with any person or resource that gives a unique advantage to some students. For example: participating in private group chats, posting questions and reading solutions on websites, hiring or requesting external help. All of these would give some students advantages that would not be available to other students.
Common Questions:
 What resources can I use on assignments? You can use your notes, the textbooks, online references such as videos and Wikipedia, and any online calculators that are freely available. You can also contact Parker and the TAs. If you’re unsure whether you’re allowed to use a particular resource: Ask!
 What calculator do you recommend for this course? When designing the course, and checking the assignments, I used WolframAlpha and Desmos.
 What recources are forbidden durings tests and assignments? Anything that is private or specific to you. You cannot post questions online to websites such as Chegg and other studyhelp sites, participate in group chats, ask for help from online communities. Please note: You can ask the instructional team for help?
 How does the instructional team monitor academic integrity? We grade the assignments and tests. If we spot suspicious answers, we enter them in to a spreadsheet. We also look at popular study help websites to see if our the material from our course is being posted online. At the end of the course, we compile all the academic integrity material and send it to the Office of Academic Integrity.
 What should I do if someone is pressuring me in to cheating? Contact Parker. Send any screenshots / photos / text messages that are relevant. You have the right to succeed in this course without other people trapping you in an academic integrity violation case.
References on Academic Integrity
Campus Resources
Facilitated Study Groups
Facilitated Study Groups (FSGs) are weekly dropin collaborative learning sessions for students who want to improve their understanding of challenging content in selected courses at UTSC. FSG sessions give you a chance to discuss the lecture material and important concepts, develop study strategies and fresh approaches, and work through problems as a group to prepare for your assignments and tests.
Research shows that students who regularly attend FSGs gain a deeper understanding of the material and, on average, achieve better grades. It’s also a great way to meet classmates and study in a relaxed, judgmentfree space.
Center for Teaching and Learning Academic Learning Support
The Centre for Teaching and Learning provides academic learning support to students through online tutoring, workshops, and peer supports to drive student success. To find out more about all their offerings, see this website:
https://uoft.me/AcademicLearningSupport
Math and Stats Support
The Center for Teaching and Learning’s Math & Statistics Support provides free seminars, workshops, virtual tutoring, individual appointments, and smallgroup consultations to improve students’ proficiency in various subjects of mathematics and statistics. Their main goal is to create a friendly, vibrant environment in which all students can come to learn and succeed.
For their online help offerings see: https://uoft.me/MathStats