MAT 240: Algebra I, Fall 2023
Instructor: Florian Herzig;
my last name at math dot toronto dot edu
Lectures: Tuesdays 11-1pm, Thursdays 12-1pm (in person)
Office Hours (online/zoom): see quercus
Official syllabus
TAs:
TA Office Hours: see quercus
Shortcuts:
- About this course.
- Sections covered and lecture notes.
- Calendar:
Week 1, Week 2,
Week 3, Week 4,
Week 5, Week 6,
Week 7, Week 8,
Week 9, Week 10,
Week 11, Week 12.
About this course
Pre-requisites: High school level calculus
Co-requisites: MAT157Y1
Textbook:
Friedberg, Insel and Spence, Linear Algebra, 4th or 5th edition, Prentice Hall.
Course description:
This course is an introduction to linear algebra over an arbitrary field aimed at students with a very serious interest in mathematics. Linear algebra is a subject that is fundamental and very useful in pure as well as applied mathematics. The emphasis in this course will be on the conceptual structure of the subject, building up from basic axioms using proofs. You will also learn about some computational techniques.
Roughly, we will cover the first four chapters of the textbook.
Assignments:
There will be weekly assignments, which you must submit by the due date. No late assignments will be accepted. Assignments will carry equal weight, and the lowest two assignment scores will be dropped.
We intend to collect your assignments using Gradescope (gradescope.CA, not .COM!). Your @mail.utoronto.ca account will be automatically signed up for a free account when you are assigned your first problem set. See their help centre for technical support.
It is important that your solutions are legible (please preview to verify). It may take a bit of time to get used to Gradescope, and sometimes small technical issues occur. Please make sure to upload well ahead of the deadline to avoid any problems.
You may discuss your homework with classmates, but you have to write your solutions on your own, in your own words. Otherwise, it is considered unauthorized aid or assistance (working too closely with another student on an individual assignment so that the end result is too similar), which is an academic offence under the University's Code of Behaviour on Academic Matters. If you find the solutions in books or on the internet, you must quote your source and still write it up in your own words! Otherwise, it may count as plagiarism, which again is an academic offense.
Term test: Oct 31, 11:10am-1pm (in person).
No aids (including calculators, phones, smart watches, etc.) will be allowed. More details about the test will be provided as the time approaches.
If you miss the term term for illness or another valid reason, you must submit the official U of T absence declaration within one week of the test, or your test mark will be counted as 0. If you provide a valid reason within one week, the marking scheme will be adjusted appropriately. There will be no makeup term test!
Final assessment:
This will take place in person during the December final assessment period and will be 3 hours long.
It will be on December 14.
How to do well in this course:
Attending class and tutorial consistently will not be enough: it is important to keep up with the material and avoid falling behind.
This course may be a lot more abstract and fast-moving than the mathematics courses you have taken before.
Therefore, you should spend enough time, at least several hours every week, on reading and gaining practice by doing problems (homework or further problems).
Working on problems (both proof questions and calculations) is a good way to test yourself if you are understanding the material.
Please also note that (1) this course is not for everyone, and there is another linear algebra course MAT223 that may be a better fit for you, (2) memorization will not get you very far.
Marking scheme:
- Homework: 20%
- Term test: 30%
- Final assessment: 50%
Academic integrity:
-
Please familiarise yourself with the University of Toronto Code of Behaviour on Academic Matters. See also this simplified version.
-
The University of Toronto treats cases of academic misconduct very seriously. All suspected cases of academic dishonesty will be investigated following the procedures outlined in the Code. The consequences for academic misconduct can be severe, including a failure in the course and a notation on your transcript. Every year, students get expelled permanently for academic offences.
Accessibility:
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University of Toronto is committed to accessibility. If you require accommodations, or have any accessibility concerns about
the course, please contact Accessibility Services as soon as possible.
Sections covered (tentative plan)
- Background (Appendices A-D, as well as the fields 𝔽p)
- Vector spaces (Sections 1.1-1.6)
- Linear transformations (Sections 2.1-2.6)
- Matrix operations and systems of linear equations (Sections 3.1-3.4)
- Determinants (Sections 4.1-4.4)
Lecture notes by Lawrence Lin
Week 1 (starting Sep 7)
- Reading: Appendices A-D: sets, functions, fields (these are important background for the whole course, before we can really start with linear algebra)
- Homework 1: I posted this on gradescope on Wed Sep 13, due Wed Sep 20. At that point you will receive an e-mail from gradescope.ca. You do not need to register for now! (If you are waitlisted or enrolled late, feel free to e-mail me for a copy of the assignment. If you enrolled late I plan to invite you to gradescope by Tuesday night.)
- You are encouraged to use LaTeX for typing homeworks (this will come in useful for other courses / projects). There is a learning curve!
It's probably easiest to get a free account at Overleaf.
Here are some links that may be helpful: Learn LaTeX in 30 minutes,
Quick LaTeX Guide (by Dana Ernst),
How to LaTeX (by Adam Blank).
- The following videos on logic, notation, definitions, and proofs (for MAT137) may be useful.
Week 2 (starting Sep 14)
- Reading: Appendix D, Sections 1.1-1.2.
- We will also discuss modular arithmetic and finite fields 𝔽p. Please see
note 1 (by Keith Conrad),
note 2 (by David Vogan) (and ignore the discussion of vector spaces in note 2 for now).
More will be done in tutorials.
- Homework 1 is due Wed Sep 20 at 11:59pm on gradescope. No late assignments will be accepted. If you didn't receive a gradescope invitation, let me know. Homework 2 will be posted Wed Sep 20.
Week 3 (starting Sep 21)
- Reading: Sections 1.2-1.3.
- Tutorials are starting September 21. Tutorial problems are now posted on quercus. You are encouraged to think about them beforehand.
- Homework 2 is due Wed Sep 27 at 11:59pm on gradescope. No late assignments will be accepted. If you didn't receive a gradescope invitation, let me know. Homework 3 will be posted Wed Sep 27.
Week 4 (starting Sep 28)
- Reading: Sections 1.3-1.5.
- Homework 3 is due Wed Oct 4 at 11:59pm on gradescope. Homework 4 will be posted Wed Oct 4.
Week 5 (starting Oct 5)
- Reading: Sections 1.6.
- Homework 4 is due Wed Oct 11 at 11:59pm on gradescope. Homework 5 will be posted Wed Oct 11.
Week 6 (starting Oct 12)
- Reading: Sections 2.1-2.2; also the notion of isomorphism from Section 2.4.
- Reminder: I will be away the week of Oct 16. Eckhard Meinrenken will teach instead of me.
- Main things covered Oct 17: linear maps, null space=kernel, range=image, nullity, rank, dimension theorem (=rank-nullity theorem=thm 2.3), thm 2.4, isomorphism
- Homework 5 is due Wed Oct 18 at 11:59pm on gradescope. Homework 6 will be posted Wed Oct 18.
Week 7 (starting Oct 19)
- Reading: Sections 2.2-2.3; also the notion of isomorphism from Section 2.4.
- Reminder: I will be away the week of Oct 16. Eckhard Meinrenken will teach instead of me.
- Main things covered Oct 19: more on isomorphisms (thm 2.19; thm 2.5), the vector space L(V,W) of linear maps
- Homework 6 is due Wed Oct 25 at 11:59pm on gradescope. No homework will be posted due to the Term Test Oct 31!
Week 8 (starting Oct 26)
- Reading: Section 2.3 (composition and matrix multiplication)
- Reminder: Term Test on Oct 31, 11:15-1:00 (during class time). More info on quercus.
- Main things covered Oct 19: more on isomorphisms (thm 2.19; thm 2.5), the vector space L(V,W) of linear maps
Week 9 (starting Nov 2)
- Reading: Section 2.3-2.6 (we covered duals, but skip double duals for time reasons)
- Reminder: Reading Week is Nov 6-10; there will be no lectures/tutorials/office hours. I'm available by appointment.
- Make sure you are comfortable matrix multiplication, its meaning and how to calculate!
Week 10 (starting Nov 16)
- Reading: Section 3.1-3.4 (we will skip the part "An Application" in section 3.3, pages 176-179 in the 4th edition; we will leave calculating the inverse of A till next week).
- Homework 7 is due Wed Nov 22 at 11:59pm on gradescope. Homework 8 will be posted on the same day.
Week 11 (starting Nov 23)
- Reading: Section 3.3-3.4 (we will skip the part "An Application" in section 3.3, pages 176-179 in the 4th edition), also the notes on determinants I posted on quercus.
- Homework 8 is due Wed Nov 29 at 11:59pm on gradescope. Homework 9 will be posted on the same day.
Week 12 (starting Nov 30)
- Reading: the notes on determinants I posted on quercus (my discussion in class differs significantly from the book).
- Homework 9 is due Fri Dec 8 at 11:59pm on gradescope.