Welcome to the math133 course webpage! Here, you'll find info about the class and links to some helpful ressources. The page will be updated every week and it will serve as the main communication tool. If any link is broken, please let me know.

schedule

course

mtwr 8:35-10:55

McConnell engineering building, room 13

tutorial

mw 13:05-13:55

Stewart biology building, room s1/3

Please note that the courses take place in-person. Courses start officially on Monday May 2nd and end on Thursday June 2nd (see key academic dates here).

course outline

The official course overview from the math department is the following : Systems of linear equations, matrices, inverses, determinants ; geometric vectors in three dimensions, dot product, cross product, lines and planes ; introduction to vector spaces, linear dependence and independence, bases. Linear transformations. Eigenvalues and diagonalization. You can find the up-to-date course outline here.

structure of the course (tentative)

• §1 prerequisites (lectures 1,2)
• §2 integers, rationnals, reals and more (lectures 2,3)
• §3 vector spaces (lectures 3,4,5,6,7,8,9)
• §4 linear maps and matrices (lectures 10,11,12,13,14)

webwork

You can try out some applied linear algebra exercises on the webwork which is accessible via MyCourses. These exercises will not be graded, but you will receive an automated feedback.

references

books

• [Axl15] Sheldon, A. (2015). Linear algebra done right. Springer.
  Links: McGill's library
• [Nic21] Nicholson, W. K. (2021). Linear algebra with applications. Lyryx Learning Inc.
  Links: Lyryx website

The topics presented will mostly follow chapters 1,2,3,5 and 10 from Axler's book with an emphasis on examples.

additionnal ressources

• If you need some help with "§1 prerequisites", take a look at the following book:
  [Vel19] Velleman, D. (2019). How to prove it : A structured approach. Cambridge University Press.
  Links: McGill's library
• Grant Sanderson (also known under the pseudonym 3blue1brown) is a mathematician from Stanford who developped incredible expertise in math visualisation. He built a series of short videos explaining linear algebra. You can find those videos for free on Youtube here.

The math help desk is open from 12:00 to 17:00 all summer at Burnside 911. There, you can find a (free!) tutor which will answer all your questions regarding our class.

More ressources will be added in the future.

course calendar

The following calendar can give you a good impression of what's to come! Don't forget to scroll down to get some more precise info about quizzes and exams.

evaluations

There will be a 2h midterm (35%) and a 3h final (55%). There will be 2 1h quizzes (2.5% each) and one take-home quiz (5%) which students get full marks for attempting. Evaluations will all be closed book.

exams

As time goes, you will be able to find the pdfs of past exams here.

• quizzes : quiz 1, quiz 2 and quiz 3
• midterm : practice midterm, midterm
• final : practice final, final

lectures

• 02.05 Presentation of the course outline. §0 Why you should care about linear algebra : Key examples of solving a system of two linear equations with two unknowns. §1 Prerequisites. 1.1 Basic logic : "and", "or", "implies", "is equivalent to", "for all" and "exists". 1.2 Basic set theory : Sets, membership, equality, subset. Functions between sets. Injective, surjective and bijective maps. The preimage of a subset. Cartesian products.
• 03.05 §1 Prerequisites : cardinality of a finite set. §2 Integers, rationnals, reals and more : Definition of rationnal numbers. Irrationnality of $\sqrt{2}$. Construction of $\mathbb{R}$ via decimal expansions. Algebraic properties of $\mathbb{Q}$ and $\mathbb{R}$. Definition of complex numbers. Definition of the norm and complex conjugate of a complex number.
• 04.05 §2 Integers, rationnals, reals and more (continued): Definition of quaternions. Definition of the norm and conjugate of a quaternion. The multiplicative inverse of a quaternion. §3 Vector spaces : $\mathbb{R}^2$ and it's properties. Definition of a vector space.
  As a sidenote, you can find the wiki page of Hamilton (who discovered the quaternions) here. He's also responsible for Hamiltonian mechanics.
• 05.05 Quiz 1, pdf available here.

tutorials

• 02.05 No tutorial
• 04.05 Tutorial 1 problems can be found here.

office hours

I will be holding office hours Wednesday the 18th at Burnisde Hall 1033 starting at 14:05 and ending at 16:00.

lectures

• 16.05 §3 Vector spaces (continued): Linear dependence lemma, size of linearly independent list vs spanning set. Definition of a basis and of the dimension of a vector space. Reduction of spanning sets to a basis. Computation of the dimension of $\mathbb{F}^n$, of $\mathbb{C}$ and of $\mathbb{H}$.
• 17.05 §3 Vector spaces (continued): Computation of the dimension of the subspace of polynomials of bounded degree. Extension of linearly independent list to a basis. Subspaces and their dimension. Computation of the dimension of subspaces of $\mathbb{F}^n$ defined by one linear equation. End of the material covered on the midterm.
• 18.05 §3 Vector spaces (continued): Vector geometry. Norm and scalar product of vectors in $\mathbb{R}^n$. Cauchy-Schwartz inequality and the triangle inequality. Equations defining planes and lines in $\mathbb{R}^3$.
• 19.05 Midterm, pdf available here and the solutions can be found here.

tutorials

• 16.05 Tutorial 4 problems can be found here.
• 18.05 Practice midterm can be found here and the solutions can be found here.

important info

I will be holding office hours Monday the 30th at Burnisde Hall 1033 starting at 14:05 and ending at 16:00.

Practice final can be found here and solutions can be found here.

lectures

• 30.05 §4 Linear maps and matrices (continued) : Invertible linear maps and bijective linear maps. Invertible matrices. Determinant of a matrix. Properties of the determinant.
• 31.05 §4 Linear maps and matrices (continued) : Change-of-basis matrix. Determinant of a linear map. End of the material covered on the final. §5 Eigenvalues and eigenvectors : Invariant subspace decomposition. Diagonalization of a matrix. Diagonalization depends on the base field.
• 01.06 Final exam, pdf available here and solutions available here.

tutorials

• 30.05 Tutorial 6 problems available here.
• 01.06 no tutorial, final exam day