MAT240F Algebra I
Classes: T111, R12;
Room: HS610. (Health Sciences, 155 College Street).
Midterm exam: T111 on October 30 , EX 100 (Exam Centre)  note change of date.
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 1112: November 26  December 4, 2018

Tuesday, Dec 4 will be our last class. There won't be tutorials during the week of Dec 3, but we will have offices hours, offered by the TA's and myself. Details TBA.
 The final exam is on Tuesday, December 11, 25 pm. The official source for time, location, etc is the Arts and Sciences webpage. Be sure to go to the correct room (depending on your last name). I'll give some more information later, but already one warning: The exam has `no tools allowed'. Do not bring calculators  before you know it, it will count as an academic offense, even if we all know that they are useless!

I've posted previous exams on Quercus.
Required reading: We will be treating determinants a bit different from the textbook. An online reference along similar lines is DentonWaldron:
p.96p.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 multilinear 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).
 Section 4.2, problems 1, 2, 5, 7, 9, 11. Also compute the last few determinants using the method explained on p. 234 (using that the determinant of an upper/lower triangular matrix is the product of diagonal entries).
 Section 4.3, problem 22. (We've encountered the Vandermonde matrix as the changeofbasis matrix from the standard basis of polynomials to the basis given by Lagrange polynomials. Now it's time to calculate its determinant.)
 Section 4.3, problem 28, Section 4.4, problem 1, Section 4.5, problems 310
Week 0: September 67, 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, 1:302:30 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 614, 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:
 Appendic C. Covered in detail. In class, we also talked about the field
$\mathbb{Z}_p$ for p prime (and the fact that $\mathbb{Z}_k$ is not a field when k is not prime).
A reference for some of this material are these
notes
by David Vogan, which also talks about the field with 4 elements.
(For now, skip the paragraph involving vector spaces. Also, note that
$\mathbb{Z}/k\mathbb{Z}$ is just a different notation for what we denoted $\mathbb{Z}_k$.
)
 Appendix D. Will be covered in detail next week, but without a proof of the fundamental theorem of algebra.
Homework #1.
The email link from Crowdmark, containing your first handin homework assignment will be sent out this Friday, September 14, before midnight. (You won't find it posted elsewhere!) IMPORTANT: Do not share your link with others, it's your personal link. If, for some reason, you do not receive this email you can ask me to resend it to you. (But please check your spam folder first.)
The `help' button in the lower left corner has various instructions. In particular, there is are instructions regarding
`completing and submitting an
assignment'. As you will see from the instructions given by crowdmark, you can handwrite the solutions (use a different page for each problem), and then scan them
or take a picture to create a pdf or jpg file for uploading. (A separate set of pages for each problem.) Scanning is available for
free
at many UofT libraries. Also, I have learned that there are excellent
apps for smartphones to scan documents, generating a pdf that's much
better quality than a jpg picture. Make sure that whatever you upload is readable  you can use the preview option on crowdmark for that. According to the crowdmark instructions, you can doublecheck and resubmit anytime before
the due date.
Alternatively,
you are very much encouraged to type the solutions using LaTeX .
In case
you're unfamiliar with LaTeX, here is a
pdf document
with some instructions, produced using LaTeX,
and here is its
source file . You can rightclick to
downlad the source file, rename it (just make sure it ends on .tex), and
modify it.
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):

Prove in detail that $\mathbb{Z}_k$ is a field if and only if
$k$ is a prime number. Hint: 1) Show first that if $k$ is not prime, then $\mathbb{Z}_k$ is not a field. (In fact, if $l$ is a natural number between $1$ and $k$, and $l$ divides $k$, show that $[l]$ has no multiplicative inverse.
2) (This part is harder.) Next, if $k$ is prime, show that for any given $l$ between $0$ and $l$ the numbers $[l],[2l],[3l],\ldots,[kl]$ are all distinct. Hence, one of them is equal to $[1]$. Use this to show that every nonzero element has a multiplicative inverse. (Actually, I may cover this in class.)

Let $P$ be the set of polynomial functions $p\colon \mathbb{R}\to \mathbb{R},\ \ x\mapsto p(x)$. Define
addition and multiplication of polynomials by adding or multiplying the values
pointwise. Is $P$ a field?

Let $R$ be the set of rational functions on a real variable $x$. The elements are thus functions $f$ that can be written as quotients of polynomials, e.g.
$$ f(x)=\frac{1+x^2}{12xx^4}.$$ The domain of definition of $f$ is the set of
all $x$ for which the denominator is nonzero. We identify two such functions if they coincide on their domain of definition, for example, $\frac{x}{x}=1$.
Is this $R$ a field?
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 1721, 2018.
Tutorials: The tutorials start this
week.
Required reading: Reread Appendices AD, 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):
 Do examples with complex numbers! Problems with solutions can be found
on the web, for example
here
and many other places.
 Section 1.1: 3; Section 1.2: 1, 12, 18, 21;
What's wrong with the calculation
$$ 1=\sqrt{1}=\sqrt{(1)(1)}=\sqrt{1}\sqrt{1}=i\cdot i=1\ \ \ \ \ ??\ \ \ $$
Week 3: September 2428, 2018.
As announced in class: The date for the midterm exam has been moved to October 30, 2018.
On Thursday, we will have a 30minute quiz, in class (between 12:301:00). 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: Reread 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):
 Do examples with long division of polynomials, finding roots.
 Show that $\mathbb{Z}_2$ is a subfield of the field with $4$ elements.
 The characteristic of a finite field $F$ is the smallest number $p$ such
that $1+1+\ldots+1=0$ (with $p$ summands). Show that the characteristic is a prime number, and $\mathbb{Z}_p$ is a subfield of $F$.
Week 4: October 15, 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:
 Q: Should I write my name and student number on the submitted pages? A: It is not necessary; Crowdmark links the submisson to your email address. In fact, we prefer if not, because the grader won't know who you are.
 Q: I didn't receive my Crowdmark email, or lost the link. What to do? A: Send me an email; I can resend your personal link.

Q: I have trouble uploading, what to do? A: If absolutely necessary, as a last resort, send
me your work as an email attachment.
Prove the `basic properties' of vector spaces mentioned in class:
 $x+y=x'+y\ \ \Rightarrow\ \ x=x'$,
 $a x=a x',\ a\neq 0\ \ \Rightarrow\ \
x=x'$
 $a x= a' x,\ x\neq 0 \ \Rightarrow\ \
a=a'$
 $0\cdot x=0$ for all $x\in V$,
 $ax=0\ \ \Rightarrow\ \ a=0 \mbox{ or } x=0$
The empty set is not a vector space. Why?
Additional homework (do not hand in).

Section 1.3: 1, 19, 20, 22, 31.

Section 1.4, Problems 2,3,12,13,14,15. (At this point
we won't cover much along the lines of problem 2,3, but we'll do it soon, more systematically. It's good to get some practice with this now .)
Week 5: October 812, 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).
 Section 1.5, Problems 1, 2, 3, 9, 10, 15, 18.
 Let $V$ be the vector space over $F$, consisting of all infinite sequences
$(a_1,a_2,\ldots)$ with $a_i\in F$. Let
\[ S=\{(1,0,0,0,\ldots),\ (0,1,0,0,\ldots),\ (0,0,1,0,\ldots),\ \ldots\}.\]
Show that $S$ is linearly independent.
What is $\operatorname{span}(S)$? Can you find another linearly independent subset
$T\subset V$, also with infinitely many elements, and such that $\operatorname{span}(S)\cap \operatorname{span}(T)=\{0\}$ In fact, such that $S\cup T$ is still linearly independent?
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 1519, 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. Ranknullity Theorem (=Dimension Theorem).
Additional homework (do not hand in).
 Section 1.6, Problems 1, 7, 9, 10
 Section 2.1, Problems 1, 2, 3, 21, 24, 25, 26, 27

Let $F$ be a field, and $X$ a finite set. Show that the set of functions
$\mathcal{F}(X,F)$, with addition and scalar multiplication defined `pointwise', is a vector space, and describe a basis of this vector space.
More generally,
if $V$ is a vector space over $F$, show that $\mathcal{F}(X,V)$ is naturally a vector space, and that a basis of $V$ determines a basis of this vector space.
(Hint: Write $X=\{c_1,\ldots,c_n\}$ and follow the strategy from the Lagrange Interpolation Theorem.) What happens if $X$ is an infinite set?

Did you do the `just for fun' problem from last week? To repeat:
Just for fun: A famous linear algebra puzzle: 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.
Here is the Solution.
Week 7: October 2226, 2018
Required reading: Chapter 2.22.4
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', `onetoone', `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:
 The midterm test takes place on Tuesday, October 30, 11:101:00 in the
Examination Centre, EX100.
 Please bring your student ID !!
 It's `no tools allowed'  especially no electronic gadgets.
 It's a `crowdmark' exams, which means that we'll scan your solutions,
and grade and return online. It is hence ok (perhaps even recommendable) to use a pencil, but if so, please use a dark pencil since otherwise it won't come through on the scan.
 We will bring some scratch paper, for rough work. However: Do not hand in scratch
paper. Only pages with a bar code will be graded.
 Material covered: Anything covered in this course until including this Thursday. (Thus, up to `matrix of a linear transformation'.)
 See Quercus, under Course Materials, for last year's midterm. (Sofar, without solutions.)
 There will be additional TA office hours before the test; for
information see the announcement on Quercus.
Week 8: October 29  November 2, 2018
The Midterm exam took place on October 30. Rough solutions
are posted on quercus.
 Required reading: Chapter 2.22.4

Coordinate representations $[v]_\beta$ of vectors $v$ and matrix representations $[T]_\beta^\gamma$ of linear maps $T$. The matrix of a linear transformations,
matrix multiplication.
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 1216, 2018
This week's office hours will have to be moved to Thurday, Nov 15,
4:305:30.
Required reading: Section 2.52.6, Section 3.13.3
(you can skip p.176179)
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).
 Section 2.5, Problems 1,2b,2c,3c,3d,6a,7,8,10
 Section 2.6, Problems 1, 2, 4, 13, 14, 15, 16
 Section 3.1, Problems 1,2,6
 Prove that the dual space to $V=F^\infty_{fin}$ is $V^*=F^\infty$.
Show that the natural map $V\to (V^*)^*$ is not an isomorphism.
Week 10: November 1923, 2018
Required reading: Sections 3.13.4.
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 $(Ab)$. 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).
 Section 3.3, Problems 1, 2, 4, 10
 Section 3.4, Problems 1, 2 (pick a few)
 Section 4.1, problems 1, 3, 8, 9,

(1) Prove that if $A,B\in M_{m\times n}(F)$ are in reduced row echelon form, and
with $N(L_A)=N(L_B)$, then $A=B$.
 (2) Prove that the reduced row echelon form of a matrix is uniquely
determined by the matrix, and does not depend on the order of the row
operations used. Hint: Use (1) above.