MAT240F Algebra I

Classes: T11-1, R12; Room: HS610. (Health Sciences, 155 College Street).

Midterm exam: T11-1 on October 30 , EX 100 (Exam Centre) -- note change of date.

Course description

This course is an introduction to Linear Algebra, aimed at students in our specialist programs.

Text Book: Linear Algebra (fourth edition) by Friedberg, Insel and Spence, Prentice Hall. (Required reading.)

Prerequisite: High school level calculus

Corequisite: MAT157Y

Course outline

Course syllabus

The required homework problems for this course will be sent online, via Crowdmark. You will be asked to submit the solutions electronically, via Crowdmark. (More information will be given later.)

Our late policy is as follows: No late assignments will be accepted.

You can certainly discuss homework with classmates, but you have to write up the solutions yourself, 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.

Here is a sample latex source file and its pdf output for download, in case you want to type up your solutions in latex and need some hints.

The `calendar' below will be updated as the course is underway. For example, I may give a few additional problems (not to be handed in), informations on tests and exams, required readings, and additional references.


Please scroll all the way down for calendar entries from past weeks.

Week 11-12: November 26 - December 4, 2018

  • Required reading: We will be treating determinants a bit different from the textbook. An online reference along similar lines is Denton-Waldron: p.96-p.138; I will also post my notes. But you should nonetheless look at our textbook's treatment (Chapter 4) as well.

  • Material covered: Definition of determinant as a multi-linear form on column vectors, the formula as a sum over pemutations. Properties of determinants: Row and column operations, $\det(A^t)=\det(A)$ , $\det(AB)=\det(A)\det(B)$. The fact that $\det(A)\neq 0$ if and only if $A$ invertible. Computing $\det(A)$ be reducing to upper triangular form. Computing determinants by expansion along row or column. Cramer's formula. Formula for inverse matrix.
  • Additional homework (do not hand in).

  • Past weeks

    Week 0: September 6-7, 2018.

    First class on Sep 6, 2018. We talked about organizational matters, and started reviewing notions from set theory. My office hours are set for Thursdays, 1:30-2:30 or by email appointment. I'm also available right after Tuesday or Thursday lectures, outside the classroom (unless there is no class after ours). Tutorials will start the week of September 17. You may find some of the MAT137 videos on notation, logic and proofs helpful. For example, I found the following video containing a bad proof quite instructive.

    Week 1: September 6-14, 2018.

  • I've posted my notes on sets, functions, and integers mod k on Quercus. Look under `Pages'. I'll continue to post some notes there, but probably not everything..
  • Required Reading: Appendices C and D, including things not covered in class. Some comments:
  • Homework #1.
  • Some do's and dont's when writing proofs: We are ok with simple logic symbols such us $\forall,\ \exists, \Leftarrow, \Rightarrow, \Leftrightarrow$ provided that you use them properly . In case of doubt, plain words are often better. Please do not use the $\therefore$ symbol since this is very uncommon in mathematical writing. Also, some other logic symbols such as $\land$ and $\lor$ are not commonly used in math texts (outside of mathematical logic), it's better to use words. Finally, avoid the use of $\times $ for multiplications (of numbers etc) since there are too many other meanings attached to this symbol -- the cross product of vectors, the variable $x$, and so on. (Use $\cdot$ instead.)
  • Additional Homework (not to be handed in):
  • Tutorials will start the week of September 17. If you couldn't get into the tutorial of your choice, please note that neither me or our TA's deal with enrolment into tutorials, and we cannot help to `get you in'! Typically, in a couple of weeks once some students migrate to MAT223, space in tutorials will free up. The assignments and tests for this course are independent of tutorials.
  • Week 2: September 17-21, 2018.
  • Tutorials: The tutorials start this week.
  • Required reading: Re-read Appendices A-D, and start looking at Chapters 1.1 and 1.2.
  • In class, we covered: Meaning of the notation $P \Rightarrow Q$ and $P \Leftrightarrow Q$. More properties of fields, for example $ab=0 \Rightarrow\ a=0 \mbox{ or } b=0$. Simlifying notation. Some finite fields. Subfields. The field of complex numbers. Real and imaginary part, complex conjugate, absolute values, and their geometric interpretation in the complex plane.
  • Homework: Assignment #1 is due Friday 11:00pm. No late work is accepted. Assignment #2 will be released before Friday 11:00 pm.
  • Additional homework (not to be handed in):
  • What's wrong with the calculation $$ 1=\sqrt{1}=\sqrt{(-1)(-1)}=\sqrt{-1}\sqrt{-1}=i\cdot i=-1\ \ \ \ \ ??\ \ \ $$
  • Week 3: September 24-28, 2018.
  • As announced in class: The date for the midterm exam has been moved to October 30, 2018.
  • On Thursday, we will have a 30-minute quiz, in class (between 12:30-1:00). This will not count towards your grade! The purpose is to give some feeling of where you stand compared to others in class, since the October 3rd `course change date' (for changing to MAT223 without academic penalty) is getting close. The quiz will consist of 6 questions; in all but one question you don't have to give explanations, only the final answer.
  • I have posted all my lecture notes sofar on quercus, under `pages'. (The notes on complex numbers have been updated, and include this week's material.) Solutions for homework 1 are also posted there.
  • Required reading: Re-read Chapters 1.1 and 1.2.
  • Material covered this week: Properties of complex numbers, calculations with complex numbers, fundamental theorem of algebra. We will start talking about vector spaces.
  • Additional homework (not to be handed in):
    Week 4: October 1-5, 2018

  • Course Change Date: October 1, 2018, is the last day to change from MAT240 to MAT223 without `academic penalty'. Click this link for information on how to change. (Note: the change date has changed.)
  • Required reading: Sections 1.3, 1.4.

    By the way, the `required readings' really are `required'. In particular, some assignment problems may rely on material from those sections, even if it wasn't covered in the lecture.

  • Some recent FAQ's regarding the Crowdmark assignments:
  • Prove the `basic properties' of vector spaces mentioned in class:

  • The empty set is not a vector space. Why?
  • Additional homework (do not hand in).
    Week 5: October 8-12, 2018.

    I'll be away this week, Steve Kudla will be teaching the class. Below is the preliminary `agenda':

  • Required reading: Chapters 1.5, 1.6
  • Plan for he week: Linear combinations. Bases and dimension. Replacement lemma. Examples and applications.
  • From now on, we'll make use of the following notations: If $A,B$ are sets, we denote $ A\backslash B = \{x\in A|\ \ x\not\in B\}.$
  • Additional homework (do not hand in).
  • Just for fun: A famous linear algebra puzzle (clever but not easy): In a town with $n$ inhabitants, there are $N$ clubs. Each club has an odd number of members, and for any two distinct clubs there is an even number of common members. Prove that $n\ge N$. Hint: Work with the field $\mathbb{Z}_2$, and use facts about bases and dimension.
  • Week 6: October 15-19, 2018

  • Required reading: Chapter 2.1
  • Material covered: Applications of bases and dimension. The formula $ \dim(W_1+W_2)+\dim(W_1\cap W_2)=\dim(W_1)+\dim(W_2)$ for subspaces $W_1,W_2$. Lagrange interpolation. Linear transformations. Rank-nullity Theorem (=Dimension Theorem).
  • Additional homework (do not hand in).
    Week 7: October 22-26, 2018
  • Required reading: Chapter 2.2-2.4
  • Material covered this week includes: The vector space $\mathcal{L}(V,W)$ of linear maps from $V$ to $W$, and $\dim(\mathcal{L}(V,W))=\dim(V)\dim(W)$. Linear isomorphisms. The Theorem that if $\dim(V)=\dim(W)<\infty$, then the conditions `onto', `one-to-one', `isomorphism' are all equivalent, but that this is false if the dimensions are different or infinite. The Theorem that $T\in\mathcal{L}(V,W)$ is an isomorphism if and only if there exists $S\in\mathcal{L}(W,V)$ such that $S\circ T=I_V$ and $T\circ S=I_W$, and that for $\dim(V)=\dim(W)<\infty$ these two conditions are equivalent. Coordinate representations $[v]_\beta$ of vectors $v$ and matrix representations $[T]_\beta^\gamma$ of linear maps $T$.
  • Additional Homework (do not hand in): Chapter 2.2, problems 1, 2, 3ace, 9, 10, 15, 16.
  • About next week's midterm:
    Week 8: October 29 - November 2, 2018
    The Midterm exam took place on October 30. Rough solutions are posted on quercus.

    Week 8b: November 5 - November 9, 2018
    Reading week (no classes). Note: November 5 is the last day to drop F section code courses from academic record and GPA.

    Make sure you are confident with the matrix multiplications discussed last week - how it works and what it means! If you want to read ahead during this week, see below.

    Week 9: November 12-16, 2018

  • This week's office hours will have to be moved to Thurday, Nov 15, 4:30-5:30.
  • Required reading: Section 2.5-2.6, Section 3.1-3.3 (you can skip p.176-179)
  • Material covered this week: The matrix of a linear transformations, matrix multiplication, change of bases. (How does a coordinate representation of a vector and of a linear map depend on the choice of bases.) Gauss elimination method, augmented matrices, elementary row operations.
  • Additional homework (do not hand in).
    Week 10: November 19-23, 2018
  • Required reading: Sections 3.1-3.4.

  • Material covered this week: The general theory of linear systems of equations. Reduced row echelon form of a matrix. How to read off a basis of the null space and range of a matrix from its reduced row echelon form. How to determine the solution set of $Ax=b$ from the reduced row echelon form of $(A|b)$. Inverse matrices. We won't have time to discuss dual bases, dual maps and their matrices in the lecture; hence I put some such things into homework 8.
  • Additional homework (do not hand in).