This (optimistic) syllabus draft is the product of the author’s thoughts after having taught MAT 224 (Linear Algebra II) at U of T following a coordinated curriculum. Activities are included for the first third of the course. Teaching assistant hours, and the course organization between teaching staff are left outside the framework of the current draft.
The course is targeted to extend and reinforce knowledge of MAT 223 (Linear Algebra I), provide a first example of a pure mathematical abstract definition as a generalization of mathematical examples, introduce a new framework for mathematical proofs outside of calculus-like epsilon-delta reasonings, and explain the applicability of the obtained results. From a practical point of view, the main differences with MAT 223 are expected to be the following: introduction and explanation of abstract definitions, abstract proof exercises, linear algebra applications, high-level structural theorems.
Grading policy
20% (4% * 5, best 5 results) : homework 6% : peer grading 24% (8% * 3, best 3 results) : quizzes 20% : midterm 30% : final exam 7% (extra grade) : group project
Timeline
| Week | Material | Assignments | Tutorial activities | Additional activities |
|---|---|---|---|---|
| Week 1 | 3 lectures | Peer-graded HW | PBL | |
| Week 2 | 3 lectures | Quiz | PBL | |
| Week 3 | 3 lectures | HW | Practical problem solving | |
| Week 4 | 3 lectures | Quiz | PBL | |
| Week 5 | 3 lectures | Peer-graded HW | PBL | |
| Week 6 | Reading week | |||
| Week 7 | 3 lectures | Peer-graded HW, Midterm | PBL | |
| Week 8 | 3 lectures | Practical problem solving | ||
| Week 9 | 3 lectures | Peer-graded HW | Practical problem solving | |
| Week 10 | 3 lectures | Quiz | Practical problem solving | |
| Week 11 | 3 lectures | HW | PBL | |
| Week 12 | 3 lectures | Quiz | Student project presentation | |
| Final week | Final exam | Student jam |
Introduction to course
Pre-lecture reading is expected from the students.
Lectures are assumed to have a large portion dedicated to activities elaborating on readings and student questions about the material.
Peer grading has to be properly introduced with well-defined and clear rules. For peer-graded assignments there are two possibilities:
- Methodical rubric-based grading: a rubric is designed for assessing each exercise and shared with the students. The upside is a higher confidence in results, a hint to students for which mistakes to look for (which is, probably, the biggest downside as well), and, for the instructor, an easier time to prevent leniency or sabotage. The downside is that it makes grading quite procedural, whereas the main goal consists in learning to assess peer reasonings (where no guarantee of correctness is present!), and the amount of creative feedback will drop considerably.
- Open-ended (without rubric): grading is more difficult, and the grade for each problem can’t have too many possibilities (1 to 10 — bad choice because of rubric absence, a +/+-/-+/-/0 solution is better with a common rubric for all exercises). Students are expected to give written commentary during grading, explaining the mistakes if any.
- + — completely correct
- +- — small flaws in the solution
- +/2 — significant flaws in the solution
- -+ — some correct ideas present
- - — incorrect solution
- 0 — unclear/not confident
Most probably, the best approach would be to combine the two (having a limited rubric, for example, or shifting from (1) to (2) gradually in the course as students get more comfortable with reading peer’s solutions).
Each submission has to be graded by at least 3 students (perfectly, five) (a rubric-based grading method allows for less gradings). It is important to note that the workload this imposes on students is high, as this essentially means every student has to grade 3-5 submissions of the entire homework. Feedback gathering during the course (polling students) to estimate the workload that grading requires should help adapt the workload. In case the workload seems completely unfeasible, reverting to classical grading can be done immediately without any logistic complications.
Peer grading should admit a regrade request system managed by TAs and the instructor. The total weight of peer-graded assignments should not exceed 30% of the course.
An important alternative to consider is self-grading. This can be performed by providing students with a detailed rubric to be able to reflect on their submission. I personally consider that self-grading carries too much risks and provides significantly fewer benefits in terms of creating a community experience.
Problem-based learning is supposed to be an important but occasional and non-graded part of the course. It should be presented in the form of open-ended questions discussed during some tutorials with prior TA training or during lectures depending on the necessity of problem solving discussions during tutorials. During the first half of the course PBL activities are intended to motivate naturality of mathematical abstraction whereas closer to the end of the course PBL is intended to showcase some applications and further directions.
Office hours are intended in a community format: there should be a dedicated space with tables for groups of students, and the format of students coming to discuss/work on the course should be encouraged as much as interacting with the instructor currently in charge of the office hours.
A Padlet feedback board, and an anonymous feedback form are tools intended for gathering student feedback on the course components. A Piazza board is the intended medium for managing student questions.
There is an extra credit group project planned. The intended output is a poster describing an application using linear algebra in the area of choice of the students or a project revolving around one of the problem-based learning activities. The projects are intended to be assessed in a poster session format.
Each topic has 5 possible components to the syllabus:
- A catchphrase: If you would reduce the topic to one catchy sentence, what would it be?
- Targeted questions: Which high-level questions do we intend to address?
- Learning objectives: Which skills do we intend to get?
- Activities: HW, quizzes, midterm, exam, additional activities.
- PBL: Addressing a open-ended problem.
Week 1: A reminder about set theory
Catchphrase:
Sets, the main abstraction of mathematics (and humanity?)
Which questions are we targeting to answer:
- Why is the notion of a set natural to consider and use? Where do sets arise in our perception of the world?
- What can be an element of a set?
- What natural operations can be performed on sets?
Learning objectives:
- Understanding numbers (integers, reals, …) as a set
- Being able to perform set operations on explicit sets
- Being able to verify set theoretic identities using Venn diagrams
- Being able to describe common examples of sets using set builder notation
Activities:
- MAT223 revision exercises: peer-graded (the target is both to introduce peer grading and to recall some of the crucial skills from MAT 223)
PBL:
- What is a set? The intended discussion is to underline how tricky the definition of set in fact is. In particular, this discussion is intended to outline the difference between “common sense” definitions and mathematical definitions. Some classical paradoxes can be discussed.
Week 2: Examples of structures: vectors, polynomials, functions
Catchphrase:
Detecting the similarity between operation structures.
Which questions are we targeting to answer:
- What makes vectors, polynomials and functions similar to each other?
Learning objectives:
- Identifying the operation: given an operation identify from the algebraic context where is the operation defined. In particular, differentiate between addition in the field and in the vector space; differentiate between multiplication in the scalar field and multiplication of a vector by a scalar.
- Being able to interpret vector space operations on the space of functions. Given two functions being able to write the values of their sum.
- Given two sets with some (natural) operations, students should be able to determine the similarities and differences between the two.
PBL:
- Define real numbers.
Week 2: Vector spaces: a (possibly, first) example of an abstraction of abstractions
Catchphrase:
Generalize the similarity of constructions.
Which questions are we targeting to answer:
- What is the purpose of an abstract definition like the vector space one?
Learning objective:
- Ability to verify that previous examples are/are not vector spaces
PBL:
- Which easier operation abstractions would we have introduced outside of linear algebra? (numbers ⇒ group, ring, field)
- What happens when some conditions on the vector space are being removed?
Week 3: Linear independence of a family of vectors
Catchphrase:
Understanding relations between vectors.
Which questions are we targeting to answer:
- Given several vectors in the vector space, which other vectors automatically lie in the vector space because of the limitations of the definition?
- Why is it important to introduce linear independence of vectors?
Learning objectives:
- Determine if a family of vectors is linearly independent.
Activities:
- HW on examples of vector spaces and linear independence.
Week 4: Basis, span
Catchphrase:
Describing a vector space through a small set of its members.
Learning objectives:
- Determine if a vector lies in the span of a set of other vectors
- Determine if a set of vectors span a given vector space
PBL:
What is an example of a basis of the space of all real-valued functions?
Week 4: Dimension
Catchphrase:
The amount of vectors uniquely defining the vector space.
Which questions are we targeting to answer:
- Why do we have to do so much effort to define what dimension is? What do the words “well-defined” mean in the context of this definition?
- How would we check if two vector spaces have the same dimension?
Learning objectives:
- Calculate the dimension of a vector space given a spanning set
- Compare and calculate dimensions of vector spaces, subspaces, sum of vector spaces, intersection of vector spaces.
PBL:
What is dimension? What is/could be fractal dimension? What is the dimension of the space of all real-valued sequences? What about the space of all finite real-valued sequences?
Week 5: Linear transformations
Catchphrase:
Stretching, rotating, projecting: they are all the same — linear transformations.
Which questions are we targeting to answer:
- What is a linear transformation, and which examples motivate us to introduce this definition?
- How many images of vectors define the linear transformation uniquely?
- How to describe a linear transformation efficiently (through bases)?
Learning objectives:
- (MAT223): Given a geometric transformation of
or (rotation by angle , reflections, …) determine if it is a linear transformation and write its matrix with respect to two given bases. - (MAT223): Given a matrix of a linear transformation of
or interpret geometrically what does the transformation “do”. - Determine whether a specific map of abstract vector spaces is a linear transformation.
- Construct a matrix of a linear transformation of abstract vector spaces given sufficient data about it.
Activities:
- HW on dimension and linear transformations
PBL:
What other examples of transformations do you encounter/can think about? Discuss the existence of affine and projective examples.
Week 5: Kernel and image
Week 6: Composition of linear transformations, inverses, isomorphisms
Week 7: Change of basis: similarity of matrices
Week 7: Determinant
Hardest definition of the course. Defining the determinant as the only alternating multilinear function on matrices with field values.
Week 8: Eigenvalues and eigenvectors
Week 8-9: Inner products, orthogonality, Gram-Schmidt process
Week 9: Symmetric matrices, spectral theorem and SVD
Week 10: Intermezzo about fields: complex numbers
Week 10: Complex vector spaces
Week 10-11: Jordan canonical form
Week 11-12: QR, LU, Cholesky decompositions
Week 12: Applications
Netflix problem as toy example.