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

Trottier building (engtr), room 0100

tutorial

mw 13:05-13:55

Stewart biology building (stbio), room s1/3

Please note that the courses take place in-person. Courses start officially on Monday May 1st and end on Thursday June 1st (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

• §0. what is linear algebra? (lecture 1)
• §1. a zoo of algebraic structures (lectures 1,2,3,4,5,6)
• 1.1 Rational numbers
• 1.2 Real numbers
• 1.3 Lists
• 1.4 Matrices
• 1.5 End of the tour of the zoo
• §2. vector spaces (lectures 7,8,9,10,11)
• 2.1 Definitions
• 2.2 Span and linear independence
• §3. linear maps (lectures 11,12,13,14,15,16,17)
• 3.1 Definition
• 3.2 Examples
• 3.3 Operations on linear maps
• 3.4 The matrix representation of a linear map
• 3.5 The Gauss-Jordan algorithm unleashed
• 3.6 Bits and pieces

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.
  Link: McGill's library
• [Nic21] Nicholson, W. K. (2021). Linear algebra with applications. Lyryx Learning Inc.
  Link: Lyryx website
• [Str16] Strang, G. (2016). Introduction to linear algebra.Cambridge Press.
  Link: McGill's library

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 your mathematical structure, 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.
• Gilbert Strang gave in spring 2005 a series of lectures on linear algebra at MIT. These lectures were filmed and they are available for free on Youtube (34 lectures, each 50 minutes). You can find them 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.

This course was also taught in the summer of 2022. You can find the webpage here. Old quizzes and exams are available there with the solutions.

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 three quizzes (1 hour, 4% each), one midterm (2 hours, 34%) and one final exam (3 hours, 54%). Students attempting the quizzes automatically get full marks. 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, quiz 3
• midterm : practice midterm, midterm
• final : practice final, final

lectures

• 01.05 §0. What is linear algebra? Solving a system of 2 linear equations with 2 unknowns in full. §1. A zoo of algebraic structures. 1.1. Rational numbers. Properties of $\mathbb{Z}$, construction of the rational numbers using pairs of integers. Addition and multiplication of rational numbers.
• 02.05 §1. (continued) Properties satisfied by addition and multiplication of elements of $\mathbb{Q}$. Proof of the irrationnality of $\sqrt{2}$. 1.2 Real numbers. "Loose" definition of the real numbers. Properties of addition and multiplication in $\mathbb{R}$. Definition of a field. 1.3 Lists. Cartesian product of two sets, $n$-fold cartesian product. Addition in $\mathbb{K}^n$ and its properties. 1.4 Matrices. Definition of $M_{m\times n}(\mathbb{K})$, the set of $m$ by $n$ matrices with coefficients in $\mathbb{K}$. Addition in $M_{m\times n}(\mathbb{K})$.
• 03.05 §1. (continued) 1.4 Matrices (continued) Multiplication of matrices. Proof that matrix multiplication is not commutative. Associativity of matrix multiplication. The identity element for matrix multiplication. Definition of an invertible matrix. Proof that square matrices are not invertible in general. Distributivity of the matrix product over addition.
• 04.05 Quiz 1, pdf available here and solutions are available here. §1. (continued) 1.5 End of the tour of the zoo. Solving a system of linear system is equivalent to solving a matrix equation of the form $Ax=b$. What makes a such a system "easy to solve".

tutorials

• 01.05 no tutorial
• 03.05 tutorial 1

The practice midterm can be found here.

lectures

• 15.05 §2. Vector spaces. 2.1 Definitions. Subspaces of vector spaces and their properties. Examples of subspaces. 2.2 Span and linear independence. Definition and first properties.
• 16.05 §2. Vector spaces. 2.2 Span and linear independence. The linear dependence lemma. Finite dimensional vector spaces. Proof that in a finite dimensional vector space $V$, the length of a list of linearly independent vectors is less than or equal to the length of a list of vectors which span $V$. Definition of a basis. Invariance of the size of a basis. Definition of the dimension of a vector space.
• 17.05 §2. Vector spaces. 2.2 Span and linear independence. Examples of bases. Every spanning set can be reduced to a basis and every linearly independent set can be extended to a basis. Computation of the dimension of a subspace of $\mathbb{K}^n$ given by solutions to 1 homogenous linear equation. §3. Linear maps. Review of injective, surjective and bijective functions. 3.1 Definition. Definition of a linear map.
• 18.05 Midterm exam, pdf available here and solutions available here.

tutorials

• 15.05 tutorial 4
• 17.05 solutions of selected questions of the practice midterm

The practice final can be found here.

lectures

• 29.05 §3. Linear maps. 3.4 The matrix representation of a linear map. How to compute the matrix representation. Linearity of the matrix representation. Examples of computations.
• 30.05 §3. Linear maps. 3.4 The matrix representation of a linear map. The matrix representation of the composition equals the product of the matrix representations. Invertible linear maps and invertible matrices. 3.5 The Gauss-Jordan algorithm unleashed. Application of the theory to $T(x)=Ax$. The columns of $A$ span the image. Row canonical form. Proof that the number of pivots equals the dimension of the image.
• 31.05 §3. Linear maps. 3.5 The Gauss-Jordan algorithm unleashed. End of the proof that the number of pivots equals the dimension of the image. An application to prove linear independence. 3.6 Bits and pieces. Definition of the determinant of a square matrix (using the Laplace expansion). Statement of results involving the determinant. Cramer's rule. Application of Cramer's rule in the 2 by 2 case.
• 01.06 Final exam.

tutorials

• 29.05 tutorial 6
• 31.05 solutions of selected questions of the practice final