MAT 224 Parting Document

Contents

1 Topics 1

2 Meta-Topics 7

3 Tips on How to Prepare 7

4 Structure of the Exam 9

5 Other Courses 10

1 Topics
Below is a list of topics considered “fair game" for the upcoming final exam. You should be
prepared to answer technical questions on any of these topics. The exam is written in such
a way as to try to find your areas of weakness, so if you think you can squeeze by with only
superficial understanding of any of the topics, we are going to try to catch it.

1. MAT223: You’ve taken MAT223 and are assumed to have done well in it. So we expect
you to be able to handle questions a MAT223 student could solve. Period.1

2. Vector Spaces: What they are and related vocabulary. Recall that the only way we
know whether something is a real vector space is by checking the vector space ax-
ioms hold. We’ve seen lots of examples, some of which indicating that the additive
identities or additive inverse may not conform to what we might naively expect when
the definitions of vector addition and/or scalar multiplication are modified.

3. Subspaces: These are sets closed under the two algebraic operations given in a vector
space: addition and scalar multiplication.

4. Linear combinations: Given a vector space, we can look at creating new vectors by
applying our algebraic rules to vectors in the space - namely adding and scalar multi-
plying vectors. This produces linear combinations, one of the fundamental construc-
tions available to us.
1
Not only is MAT223 a formal prerequisite, it is a conceptual prerequisite. MAT223 should be viewed
simply as a special case of the topics listed below, so this shouldn’t be seen as additional content.

1
 5. Subspace Sums: We’ve overloaded the + symbol in multiple ways already (after all
+ in Rn and + in P2 (R) aren’t the same are they?). Here we go one further and use
the symbol + as a binary operation on subspaces themselves. The sum of subspaces
is a subspace and it’s the smallest subspace containing the union of the summand
subspaces.

6. Linear Independence: We learned about the fact that if a set of vectors is linearly de-
pendent then any linear combination of the vectors can be reduced to a linear combi-
nation of a subset of the vectors. So in this way, a linearly independent set of vectors
doesn’t have “redundancy". You should know the meaning of linear independence,
how to check for it, and have a sense of why it’s important.

7. Bases: As basis is simply a linearly independent spanning set. Put differently it’s a max-
imal set of linearly independent vectors or a minimal set of spanning vectors (make
sure you see why!). Choosing a basis for a vector space allows us to make concrete
the former abstractions by giving us a way to represent vectors in the space explicitly.
In a way, without having a basis I don’t feel I really understand a vector space.

8. Dimension: Since all bases in a given vector space must have the same size (prove
this!) we can tag each vector space with that number - the number of vectors in any
basis for the space. That number is called the dimension and it serves as sort of a proxy
for the “size" of a vector space, in that “larger" spaces should have larger dimensions.
Another way I like to talk about it is in terms of “information". Dimension captures
the number of degrees of freedom in a space and that’s essentially the amount of
information one can pack into it. 2

9. Basics of Linear Transformations: You’ll need to know what a linear transformation

is, how to verify if something is or isn’t one and some basic properties of linear trans-
formations. Linear transformations are those maps between two vector spaces which
preserve the algebraic structure, and in this way they allow for a type of correspon-
dence or “communication" between two vector spaces. We’ve also seen that L(V, W),
the space of linear transformations between vectors space V and vector space W, is
itself a vector space when equipped with the standard version of addition of trans-
formations and scalar multiplication. Viewing the set of maps between spaces as a
space in itself moves us one small rung up the abstraction hierarchy in the world of
abstract vector spaces3 .

10. Properties of Linear Transformations: Things like injectivity, surjectivity, bijectivity

and how to check for these.

11. Representations of Transformations: You should be completely, absolutely com-

fortable with the notation and calculation of things like [T ]βα where α, β are bases of
finite-dimensional vector spaces on which T acts. Moreover, you need to know what
2
Here I’m speaking loosely and metaphorically, of course
3
And, of course, the procedure of “vector spacing" L(V, W) is trivially replicable on that space itself. So one
can produce an infinite hierarchy of abstract spaces where each new space is made up of maps of two spaces
of the prior construction ad infinitum.

2
 things like [T (v)]β = [T ]βα [v]α mean and why. The notation takes some getting used to
but once you master it your life will be much easier.

12. Fundamental Subspaces: You need to be solid on the fundamental subspaces asso-
ciated to a linear operator T , namely the kernel ker T and the image Im(T ). How can
their dimensions be used to classify the properties of a given transformation? These
are obvious generalizations of the subspaces Nul(A) and col(A) which we encountered
in MAT223. Make sure you understand how/why.

13. The Dimension Theorem: In MAT223 we knew this theorem as the Rank-Nullity
Theorem. It is a tool which allows us to answer a lot of questions simply by appeal-
ing to numerical values. It can help to solve lots of problems, and is the main tool
relating the fundamental subspaces of operators on finite dimensional vector spaces.
The fact that the textbook devotes an entire section on tis one theorem is telling. I
likened it in my lectures to a “conservation of information" equation; make sure you
can understand how/why.

14. Compositions of Operators: Given T ∈ L(V, W), S ∈ L(U, V) you need to know what’s
meant by S T and how to represent it as a matrix, and how that representation de-
pends on the representations of S and T .

15. Inverses of Operators and Isomorphisms: For T ∈ L(V, V) you need to know what’s
meant by T −1 . Invertible linear maps are known as vector space isomorphisms and
you need to know when a given transformation is an isomorphism. We know that
finite dimensional spaces with the same dimension are isomorphic (i.e. there’s an
isomorphism between them).

16. Change of Basis: This is a foundational idea - understanding how representations

of objects depends on the basis one uses and how to relate two different represen-
tations. The change of basis matrices [I]βα provide a kind of “dictionary" linking two
different representations. Or, if you prefer, it acts as a machine transforming vectors
in the α basis to the same vectors in the β basis.

17. Similarity and Conjugacy: Representations of linear transformations in different

choices of bases are related by similarity, i.e. they will be similar matrices. Given
a square matrix A, the set of all matrices similar to A is a so-called conjugacy class.
Within the conjugacy class, similarity between matrices defines an equivalence re-
lation. Thus, the space of square matrices is partitioned into conjugacy classes. It
follows from the change of basis formulae that a linear operator’s representation is
always a member of one and only one conjugacy class. Hopefully these concepts
hold some intuitive power.

18. Determinants: You should know how to calculate determinants of n × n matrices.

You should also be comfortable with the basic properties e.g. det(AB) = det(A) det(B),
det(cA) = cn det(A), etc.

19. Eigenvalues and Eigenvectors: Given a square matrix A, you should know what
eigenvectors and eigenvalues are and how to find them. What are eigenvectors geo-
metrically? You should understand geometric and algebraic multiplicity of an eigen-

3
 value. You should also know about the characteristic equation (what it is and how to
find it) of a transformation, as well as invariant subspaces (what they are and how to
find them).
20. Eigenspaces: Make sure you know what an eigenspace is and how to find one. Are
all elements of an eigenspace an eigenvector? Is Eλ a subspace for all λ ∈ R? Are they
ever subspaces?
21. Diagonalization: You need to know what’s mean by diagonalizability, and how to
check whether a linear transformation is or isn’t diagonalizable. What are some equiv-
alent conditions for a matrix being diagonalizable? If a matrix is diagonalizable how
do you diagonalize it?
22. Polynomials: We’ve exhaustively gone over what polynomials are, and you should
not only know basic properties like the Fundamental Theorem of Algebra4 , you should
know about polynomial division and polynomial factorization, especially the connec-
tion of the splitting of polynomials has to do with the field over which they split.
23. Fields and Complex numbers: C, R, Q and others are examples of a field, which is
a formal, axiomatically defined algebraic object. Field elements we just call “num-
bers", usually. We’ve also seen some finite fields. Complex numbers, thanks to the
fundamental theorem of algebra, are a more natural space to perform arithmetic and
algebra5
24. Geometry: We’ve covered inner products on Rn together with more general inner
product spaces (like Hermitian inner product spaces). You’ll obviously need to be
aware of the Cauchy-Schwartz and triangle inequalities as well as the general Pythagorean
equality. This geometry also of course includes the idea of orthogonality, a crucial
component of the later sections.
25. Symmetry: The idea of a symmetric operator is reliant upon the particular inner
product one has. But you need to know that if, say, hT v, wi = hv, S wi holds for all v, w in
an inner product space, then S = T T . Namely, this algebraic identity uniquely defines
the transpose of an operator. Once we have the idea of transpose, the definition of
symmetry is then obvious. We care about symmetric operators for many, many rea-
sons, not the least of which is that they are the ones most commonly encountered in
applications.6
4
This theorem, which can be stated in many equivalent ways (I’ll state it as saying that the complex field
is algebraically closed) has a rich history. The first proof is often attributed to Gauss who at age 22 claimed
a complete proof. By modern standards, his proof was (fatally) incomplete and, even Gauss himself noted in
a footnote regarding his unproven assumptions used in the proof that "As far as I know, nobody has raised
any doubts about this. However, should someone demand it then I will undertake to give a proof that is not
subject to any doubt, on some other occasion". No one demanded and so he did not ever provide that "other
occasion". Gauss gave other proofs of the theorem later on. It seems the first truly rigorous proof came about
7 years afterwards and was given by a French amateur mathematician who was employed as a bookstore
manager. It’s rare for hobbyists to give profound contributions to the field.
5
But, notice that unlike R, C is not an ordered field. By this I mean C doesn’t have a natural order relation
< with the property that for all w, z ∈ C either w < z, z < w or w = z.
6
For example, the covariance matrix in statistics is symmetric. Further, given a matrix X whose rows are
examples of observed vectors, the symmetric matrix X T X is often nicer to work with than X (note that this is

4
 26. Projections: Linear operators P : V → S , where S ⊂ V which are symmetric and
satisfy P2 = P are called orthogonal projections and are sort of like the foundational
elements in the space of symmetric linear transformations. You need to know what
they are, how to prove things with them, and the role they play in the larger story of
linear operators on vector spaces.
27. Gram-Schmidt Procedure: Given a basis for a subspace of an inner product space
there’s an algorithm to build an orthonormal basis for the same subspace. That al-
gorithm is the Gram-Schmidt one. We prefer working with orthonormal bases since
coordinate expansions are much nicer than in non-orthonormal bases.
28. Hermitian Operators and Spaces: We extended inner product space to its complex
cousin, the Hermitian inner product spaces. With this we had to define the adjoint
of a transformation T : V → V, denoteed T ∗ or T † , which is similar to the usual trans-
pose. Namely, if h·, ·i is a Hermitian inner product on complex vetor space V then T ∗ is
the unique operator S satisfying hT v, wi = hv, S wi for all v, w ∈ V. Hermitian operators
are those operators for which T = T ∗ and are the complex analogue of symmetric
operators.7 In matrix form [T ]ββ = ([T ]ββ )T , i.e. the representation of the adjoint is the
conjugate transpose of the representation of T . We care a lot about Hermitian oper-
ators, since not only do they abstract the important features of symmetric operators,
but they are encountered in applications very often. 8
29. The Spectral Theorem: You should know the spectral theorem, both what it says and
how it gets applied. We encountered it in its real form, and then later in it’s complex
form, in the context of Hermitian linear operators. This theorem is one of the major
landmarks in the MAT224 vistas.
30. Vector Spaces over Fields: Having defined vector spaces in a fairly easily-abstracted
way, we looked at extending the definition so that the scalars in our vector spaces
are coming from F, where F is often just R or C. This allowed us to look at things
like the vector space Fn and Pn (F) with the “usual" definitions of addition and scalar
multiplication.
31. Triangularization: This is our first foray into abstraction. We discussed general
shapes which representations of linear operators can admit, and saw that triangu-
lar form is connected with properties of the underlying field (namely we want the
characteristic polynomial to split over the base field). A representation takes on a
triangular form if and only if there’s an increasing sequence of invariant subspaces
(see following). So the abstract decomposition of the ambient vector space into non-
interacting subspaces on which the operator acts invariantly is the high-level concept
in this section.
actually just a rescaled covariance matrix for the observations). This kind of symmetric matrix is a fundamen-
tal object in modern machine learning. Very loosely speaking, since algebra is about equality and analysis is
about inequality, this makes C a natural algebraic space but a more complex (pun intended) analytic space.
7
These generalize symmetric operators and thus include symmetric operators as a special case.
8
For example, one can take as an axiom of the natural world that physical observables are Hermitian op-
erators in an (infinite-dimensional) inner product space (called a Hilbert space). This unusual axiom has the
defect of being awkward to state but the advantage of being true and carrying lots of implications for exper-
imentalists.

5
 32. Invariant Subspaces: One of the major pursuits in advanced modern mathematics9
is the study of invariants. For us, the abstraction of vectors is vector spaces, and the
study of vector spaces involves analysis of operators on these spaces. The study of
operators is abstracted by the study of what abstract items those operators “preserve".
That’s what the invariant subspace material is about.

33. Nilpotent Operators: You should know what nilpotent operators are and basic con-
sequences of nilpotency. For instance, if N is nilpotent, what can we say about it’s
characteristic polynomial? Are nilpotent operators always triangularizable or only in
complex vector spaces? Nilpotent operators are a major component of Jordan form
so without understanding nilpotency it will be impossible to fully master Jordan form.
You need to know about the canonical form of nilpotent operators.

34. Cycle Tableau and Cyclic Subspaces: You should know how to construct cycle tableaux
for a given nilpotent operator and, given a cycle tableaux, what the corresponding
canonical form of the nilpotent operator will be.

35. Generalized Eigenspaces and Generalized Eigenvectors - Given an eigenvalue λ (namely,

a root of the characteristic polynomial) of T : V → V of multiplicity m we can define
the corresponding generalized eigenspace Kλ = ker(T − λI)m . Then (T − λI) |Kλ will be a
nilpotent operator, with index of nilpotency at most m. Further, you should be able to
see that Kλ are T -invariant subspaces. There were no assumptions about the operator
T , so this means that for arbitrary linear operators on a complex vector space we can
construct T -invariant spaces corresponding to these nilpotent operators.

36. Jordan Form: Given a linear operator on a finite dimensional complex vector space,
one can consider its associated Jordan Canonical Form, which consists of a block-
diagonal matrix, whose blocks are mi ×mi matrices of form λi I + Ni where Ni is an mi ×mi
nilpotent operator in it’s canonical form, and λi is an eigenvalue with multiplicity mi .
To calculate the canonical form you have to first find the distinct eigenvalues and their
multiplicities. Then, you construct, for each distinct eigenvalue, the nilpotent opera-
tor Ni = (T − λi I) |Kλi and it’s cycle tableau. The cycle tableau for each such operator
gives you the corresponding geometry of the relevant Jordan Block appearing in the
Jordan canonical form. The book gives several examples, as well as examples of how
to calculate a Jordan basis, namely a basis β, with respect to which [T ]ββ takes on a Jor-
dan canonical form. Jordan canonical form and nilpotent canonical form are special
cases of triangularization. And triangularization is a special case of change of basis
we’ve seen earlier. Namely, the Jordan canonical form theorem can just as well be
stated as: Given T : V → V linear operator on complex, finite dimensional vector
space V, there exists a basis β and an invertible matrix P such that

[T ]ββ = PJP−1

holds, where, in the above, J is a block diagonal matrix with the eigenvalues of T on
the diagonal and, possibly, 1’s appearing in some of the super-diagonal entries (i.e.
9
Actually, I would say this is one of the major reconceptualizations in 20th century physics as well. Modern
particle physics is often heavily motivated by invariants in the form of so-called “gauge symmetries".

6
 those in the (i, i + 1)’th entries for some i). In other words, Jordan canonical form is
just another matrix factorization.10

2 Meta-Topics
Here are things which we may have only briefly mentioned or implicitly used but for which
you are still expected to know.
1. Induction: You should be familiar with following inductive proofs as well as con-
structing your own. This was part of the material in the Appendix of the book (as-
signed as your homework for week 0 prior to the start of term). If you are rusty on it,
get unrusty quick!
2. Sets: We expect absolutely zero sloppiness and no abuse of set notation. The notation
in this course is very inflexible and you’ve had three months to master the construc-
tion of basic sets. We expect complete perfection in this regard.
3. Grammar: I tell my sections that this is a course on reading comprehension, since
much of the course is on looking at a statement about formal, abstract things and tak-
ing a careful, thoughtful approach to interpretation. Unlike in other hermeneutical
areas the complete rigidity of our notation means the things you write are read com-
pletely literally. Think about the following snippet “Let x = a1 x1 + · · · + an xn = S + X".
Read literally, it must be the case that whatever kind of object x is, S + X must be
the same kind of object. So, is S + X a vector or a set? Are the x1 , ..., xn ’s vectors or
something else? If they are vectors, in which space do they belong? What about the
a1 , ...., an ’s? Are they numbers, matrices, vectors, or something else? What you write
will be read literally and scrupulous attention will be paid to these kinds of grammat-
ical ambiguities or contradictions. Remember, the graders only know what you’re
thinking based on what you write! So if you are having the correct idea about the
problem but express it sloppily, they will not be able to follow the logic and you may
lose points.

3 Tips on How to Prepare

Here are a few thoughtlets on best practices.
1. Unfortunately, the best preparation is to have stayed on top the entire semester and
not have fallen behind at all. It is extremely hard to get back on top if you’ve fallen
behind in a course like this since the abstract material requires significant time to
absorb properly.
10
In MAT223 you’ve seen the LU factorization of matrices and, in this class we’ve seen diagonalization
and triangularizations as matrix factorizations. There are plenty more. One that comes up very often in
numerous applications is called singular value decomposition (SVD). Beyond that, there are plenty of other
common factorizations (Schur factorization, QR factorization, QS factorization, Polar decomposition to name
a few of the common ones). Breaking complicated things apart into simpler pieces is a very general idea, and
that’s what each of these factorizations achieves. Which one is best is like asking which tool is the best for
woodworking: it depends entirely on the task at hand.

7
 2. I recommend having gone through literally every single recommended problem from
the textbook. If you’ve done that, I consider you well-prepared.

3. Please get lots of sleep the night before the exam. Brains operate better when they are
well-rested.

4. I generally recommend getting aerobic exercise the day of the exam (go for a long
walk, a swim, a run, etc). Not only does this give you a nice, quiet time to think, but
it can help your body relax a bit so that if you’re prone to anxiety, your body will
naturally have a stronger defence.

5. Remember that three hours is a long time. The test is 3 hours and there’s no need to
rush. You have plenty of time to answer each question thoughtfully and demonstrate
you’ve learned the material well. This should help those who may have felt rushed
during the other tests.

6. You’ll want to be so solid on the material that you aren’t wasting lots of time on things
that a well-prepared student could answer quickly. In other words, you’ll want the
luxury of being able to think more deeply about the things for which deep thought is
beneficial as opposed to wasting time on things we expect you to go through quickly.
So make sure that any computational type of question you can get done quick. Cal-
culations should be the easiest things we can ask.

7. Don’t think that we won’t test you on topic X. We’ve created the test with the goal
of trying to find where you are weakest, and if there’s something you’re weak on we’ll
be hoping to find it. So just make sure you aren’t leaving any areas of study out.

8. Decide in advance a strategy for how to deal with true/false questions you aren’t sure
about. Since there are deductions for incorrect answers, and since exams can be a
high stress environment, it’s good to think through how certain you want to be in
order to guess an answer, prior to test day. Obviously, the best case scenario is that
you are confident with the T/F answers (and even better if this confidence correlates
with being right!) but, failing that, if you are, say, 70% sure of an answer, should you
put down the guess? How about 65% sure? 51% sure? Everyone will have different
thresholds and different attitudes towards risk, so there’s no best answer here. But
you shouldn’t be making ad hoc decisions during the exam, you should know how
you plan to deal with things like this beforehand.

9. I recommend11 beginning the test by taking a quick look at each question, to get a “lay
of the land". When perusing the questions, I suggest coming up with numbers for
each problem. These numbers are your estimate of the amount of time you imagine
that problem might take. When working on the test, I would recommend trying to
stick to not going over the time limits you’ve given yourself for a given problem before
moving on to the next one you haven’t solved and coming back to the one you’re stuck
on later.
11
And these really are just recommendations. I don’t have a one-size-fits-all algorithm for how to write a
final exam. These are just things that I would do if it were me writing the final.

8
 10. I also recommend working on what you feel are the easiest questions first. Notice that
the “easiest" may not mean “earliest". Perhaps you find T/F questions to be easiest. Or,
perhaps you prefer calculation style questions. Whatever your preferences, I would
suggest working on those problems first since that way you can secure a nice blanket
of points right away. Then I would suggest working on problems in order of increasing
estimated difficulty. This way you are gathering as many confident points as possible
with the least amount of time.

11. Remember, we only know what’s in your head by what makes it onto the page. And
we aren’t mind-readers. If you have an idea for connecting ideas in a proof, say, and
you aren’t clear in your explanation, then the reader cannot follow the argument. Just
be very precise and clear in the way you indicate your thoughts because otherwise
there will be ambiguities which can lead to loss of points.

4 Structure of the Exam

1. The exam has 12 questions. Many of the questions have multiple parts.

2. The exam is out of 200 points.

3. The last question is a selection of true/false questions. Unlike on the term tests you
can end up with negative points on the T/F question page. If your score is, say, −6 on
that page, points will be deducted from other problems.

4. There is an optional bonus problem at the end of the exam. It is (in my opinion) not
at all an easy question. It is worth an additional 30 points.12 Frankly, since we are not
going to be generous with partial credit on the bonus question, I recommend only
attempting it if you feel confident about your work on the main exam questions. I
don’t want people to devote tons of valuable time on this harder question in vain. It’s
there purely as a bonus and as a reward for those who were well-prepared enough
to be able to spend time working on it.

5. Some of the questions will offer a nominal amount of points for leaving the question
blank, as was done on the second midterm. Again, this is there discourage people
from writing completely aimless things in the desperate hope of getting some kind of
partial credit. The amount offered will be clearly stated before the problem itself is
stated.

6. Although the exam is done on Crowdmark, you won’t receive the Crowdmark link for
quite a while, and I NEVER13 respond to emails about the final exam, so don’t bother.
Every year I say this, and every year, without fail, some students somehow imagine
this not applying to them. It applies to you, dear reader.
12
In other words, some students could conceivably score, say, 230/200 on this final exam.
13
Like, actually, truly, I never have. And I’m not going to be breaking my streak.

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5 Other Courses
Several students have asked about follow-up courses to MAT224, and what they should take
if they enjoyed the material in this term. Here are my thoughts.

1. MAT245: This is the course I created and designed in 2017 on “Mathematical Meth-
ods in Data Science". It’s mostly taken by specialists in mathematics and computer
science. Here is the syllabus from this past term http://www.math.toronto.edu/
nhoell/MAT245/MAT245_syllabus.pdf It’s a notoriously demanding course (not for
the faint of heart!), but the students who’ve completed it have consistently rated it as
the most valuable course they took. We use a lot of advanced linear algebra exten-
sively, and cover the ways linear algebra is used in machine learning and other forms
of inference.

2. MAT301: This is a course on “Groups and Symmetries". I’ve not taught it so I don’t
know the syllabus but the material should be an extension of things we’ve done in
this course. Namely, the course should cover more general algebraic objects than
vector spaces called groups. The set of invertible n × n matrices are an example of a
group14 , called GLn which is presumably something arising in that course.

3. MAT347: This is a course on “Groups, Rings and Fields". I’ve not taught it and don’t
know the syllabus but I imagine it’s a more advanced version of MAT301. We’ve seen
examples of fields in this course, and (without calling them groups) we’ve seen ex-
amples of groups as well. We haven’t discussed rings, and we also haven’t discussed
the ways these abstract things are all related. This course, I presume, covers these
algebraic objects in detail.

4. MAT401: This course is on “Polynomial Equations and Fields". It covers topics relating
to polynomials15 which we simply took for granted in MAT224. It covers the relation-
ships between polynomials and the number fields over which they have solutions.

5. APM421: This course is on “Mathematical Foundation of Quantum Mechanics". The

formalism behind atomic physics is, essentially, linear operators on (infinite dimen-
sional) vector spaces. I’ve not taught the course and don’t know the syllabus, but it
certainly must cover the basics of matrix algebras and their representations on state
spaces in quantum mechanics. You would certainly encounter more generalized ver-
sions of what we’ve discussed in MAT224, especially in that you’d be working with
linear operators on infinite dimensional spaces.

6. Any Course in Partial Differential Equations: There are several options here and
I’m not going to list them. The more mathematical options may introduce you to the
infinite-dimensional version of the spectral theorem. Even if they don’t you’d be ex-
posed to adjoint theory and infinite-dimensional inner product spaces. The prereq-
uisites for courses like this are generally MAT244 or its equivalent. I’ve taught courses
14
This is only one simple example, groups are very general and complicated. But, invertible matrices with
determinant one 1, say, is a very common and extremely important example of a group with deep applications
in physics.
15
Like the fundamental theorem of algebra.

10
 like this many times and we tend to cover quite a bit of material which extends stuff
we’ve done in MAT224.

7. MAT436: This is a course on “Linear Operators" which I think is a cross-listed gradu-

ate course. I think it’s a demanding course. But, it is also the most obvious one on the
list as candidate for “more advanced version of MAT224". Basically, working out the-
orems on infinite-dimensional vector spaces is called functional analysis and I think
this is UofT’s course on functional analysis. If you like MAT224, and you have taken
some kinds of , δ advanced analysis, this might be a course to consider. But I believe
it’s taught at a high level.

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