Math 223 Section 101
Linear Algebra
Online Course Material 

NEWS

Our final exam is scheduled for Thursday December 15 at 3:30 until 6:30 (I schedule 3 hour exams). The room is IBLC 261 (Ike Barber Learning Centre). I have posted below two old sample exams.

This is an Honours level course that both covers the material of MATH 221 and MATH 152 but adds some theoretical material. The grading standards (for me) are roughly those of MATH 221 and so some judicious scaling will be used. It will be substantially more work than MATH 221 but hopefully much more interesting.

Please note that this course cannot be taken for credit if you have already taken MATH 221 or MATH 152.


The first midterm is October 7. You will get a break with Thanksgiving.
  • Sample midterm 1 I will not post solutions but you can ask me about your solutions during office hours.  
  • The second midterm is November 9.
  • Sample midterm 2 I will not post solutions but you can ask me about your solutions during office hours.  


  • The following is a sample midterm 2 from MATH 221 from a previous year. I post it to show you how far you have come.
  • Old 221 midterm 2  
  • 2008 Sample Final Exam This sample exam is a study aid. It may be too early for you to use it profitably. I will NOT post solutions. You can come and show me your solutions and ideas if you want feedback. Reading solutions (unless you have done the problems) is not a valuable study aid in my opinion.  
  • 2009 Sample Final Exam This sample exam is a study aid. It may be too early for you to use it profitably. I will NOT post solutions. You can come and show me your solutions and ideas if you want feedback. Reading solutions (unless you have done the problems) is not a valuable study aid in my opinion.  
  • I will be putting many course materials on the web in pdf format to supplement the text. The text has some excellent explanations but note that the problems tend to be much too elementary.
    The notes look authoritative when typed so be wary. They may look perfect but may still contain errors! I don't have an editor.
    I arrive most days by 9:00. I will typically be available after class in my office MATH ANNEX 1114 at 11 each day. I will have an office hour 4:30-5:30 on Thursdays in MATH ANNEX 1118 (near my office) and imagine many assignments will be due Fridays. Other office hours will be available. I do not reliably read my email from home (i.e. evenings and weekends) nor every hour at work.

  • Course Outline: grading scheme etc. 
  • Assignment 1. Due Friday September 16 Assignment 1 solutions.  
  • Assignment 2. Due Friday September 23 Assignment 2 solutions.  
  • Assignment 3. Due Friday September 30 Assignment 3 solutions.  
  • Assignment 4. Due Wednesday October 5 Assignment 4 solutions.  
  • Assignment 5. Due Friday October 21 Assignment 5 solutions.  
  • Putnam problem from 2013 (question 11 on assignment 5).
  • Assignment 6. Due Friday October 28 Assignment 6 solutions.  
  • Assignment 7. Due Friday November 4 but will be accepted on Monday Nov 7. Assignment 7 solutions.  
  • Assignment 8. Due Friday November 18. Assignment 8 solutions.  
  • Assignment 9. Due Friday November 25. Assignment 9 solutions.  
  • Midterm 1 Solutions.  
  • Midterm 2 Solutions.  


  • Various Notes. Often for material not in text or at least not in the order we follow.

  • 2x2 matrices from first lecture  
  • 2x2 matrices from second lecture  
  • Putnam problem from second lecture  
  • 2x2 matrices as geometric transformations and also Linear Transformations  
  • Different levels of generality.  
  • Bird populations and eigenvalues and eigenvectors.  
  • White and Blue Coordinates.  
  • Fibonacci numbers.  
  • Gaussian Elimination.  
  • Determinants.  
  • Partial Fractions.  
  • Interchanging rows 1 and 2 changes determinant by -1.  
  • Axioms for a Field and for a Vector Space.  
  • Introduction to Vector Spaces including the idea that: a vector is not a vector.  
  • Ideas of Linear Independence, spanning sets and dimension. Big important topics of MATH 223  
  • Wronskian, an alternative approach to deciding if functions are linearly independent (there are other easier ways as well).  
  • Row Space and Column Space and Rank More Big important topics of MATH 223  
  • Change of Basis. You can interpret matrices in many ways and (typically) can encode a linear transformation by many matrices.  
  • Systems of Differential Equations. The basics for a homogenous system including how to proceed when there are complex eigenvalues.  
  • Complex Numbers I. Complex Numbers II. Some helpful facts about complex numbers.  
  • Some non-diagonalizable matrices and the simple matrices which they are similar to. This is not directly testable although the ideas contained are standard linear algebra and testable.  
  • Smolensky proof of the Sauer, Perles and Shelah, Vapnik and Chervonekis bound. This was called a trip to the Opera, namely a cultural field trip but not part of our curriculum.  
  • Vector Geometry. It is amazing that the dot product is so helpful for vectors. We also discuss general inner products later.  
  • Orthogonal Vector Spaces. We give the main result in terms of a general inner product.  
  • Equiangular lines. This example is reminiscent of the Smolensky proof in that dimension arguments yields a surprisingly useful bound.  
  • Gram-Schmidt process that yields an orthonormal basis.  
  • Orthogonal Projections  
  • Symmetric Matrices are Orthogonally Diagonalizable  
  • Some problems for the Petersen Graph  
  • Quadratic Forms and Conic Sections Somewhat preliminary notes.  
  • Some pictures from plotting 100 points that are relatively evenly spaced on a sphere Back points, Front points, All points. The idea is how to obtain the rotation matrix by 10 degrees around an arbitrary axis.  
  • A list of topics. May be helpful when reviewing to remind you of some of the ideas/concepts presented in class.  


  • The Math Learning Centre (or MLC for short) is a space for undergraduate students to study math together, with friendly support from tutors, who are graduate and undergraduate students in the math department. The MLC is located at LSK301 and LSK302 and is open 5 days a week 11am-6pm starting September 14th. Every undergraduate student studying Math is welcome here! In the MLC, students may join the study groups if students wish so. Please note that while students are encouraged to seek help with homework, the MLC is not a place to check answers or receive solutions, rather, our aim is to aid students in becoming better learners and to develop critical thinking in a mathematical setting. For additional information please visit our website website. .

  • Matrix multiplication and some possible interpretations other than as linear transformations. Multiplication on the left is a row operation, multiplication on the right is a column operation etc. We discuss other interpretations later in the course.  
  • An interactive 2x2 matrix multiplication: A java program that allows you to move input point A around and see the output after a 2x2 matrix multiplication. The matrix entries are a,b,c,d and the java program gives you sliders to change their values if you wish. The initial data for a,b,c,d corresponds to the matrix for the fibonacci numbers. This demo was created with GeoGebra in about 1 hour with no knowledge of Java except the idea of jar files. You can play with this free software at www.geogebra.org (NOT www.geogebra.com)  


  • Singing (NOT ME) about invertibility for you enjoyment.

    How to slice a bagel Use edible ink.

    Strange facts Any other facts you wish to add?.

    Infinite Jokes from Chris Ryan . Any other jokes you wish to add?.