Math 223 Section 101
Linear Algebra
Online Course Material 

NEWS November 2021

Our final exam is scheduled for Wednesday 8:30am December 15 in LIFE 2302 (a nice room in the old Student Union Building, from the photos of the room it appears each of you may have your own desk). Our exam is a 3 hour exam. I have arranged MATH ANNEX 1110 for December 14 from 10-12. And I will be available at some other times Dec 13,14.

advice for final Some students ask about how to study. One recommendation is to create some summary notes from what we have covered. Then reading assignments and your midterms come next. Try to figure out what you do wrong. I wonÕt entertain regrading at this point. Then you have two practice exams. Symmetric matrices should play a role.


linear transformations (representation as matrices, matrix multiplication as function composition)

Gaussian elimination including vector parametric form.

Determinants (how to calculate, product rule)

Vector Spaces (linear independence, linear dependence, basis, dimension, extending a set of vectors to a basis)

Row Space, Column Space, Rank, Nullspace, nullity thm (how multiplication on left or right affects row and column spaces)

Change of Basis

Eigenvectors/Eigenvalues (eigenspace for lambda is nullspace(A-lambda I), dimension of eigenspace is at most multiplicity as a root in det(A-lambda I), similarity e.g. A similar to B, then det(A-lambda I)=det(B-lambda I), matrix is diagonalizable if sum of dimensions of eigenspaces in equal to size of matrix or each eigenspace dimension is equal to multiplicity as a root, examples of non diagonalizable matrices, applications of diagonalizability to computing powers etc, dominant eigenvectors/eigenvalues )

Differential equations (diagonalization, exponentiation of matrices, complex numbers)

Vector geometry (length of vector, dot product and angle between vectors, projections, Gram Schmidt process, orthogonal vectors, orthonormal bases)

Orthogonal vector spaces (orthogonal complement, projection onto a vector space using an orthonormal basis for the space)

Symmetric matrices (are diagonalizable, have real eigenvalues, have an orthonormal basis of eigenvectors, Hermitian definition)

Matrices for various linear transformations including rotation about an axis in 3-space We also has lectures on applications which would not be directly testable (Proof of partial fractions idea, Smolensky proof of bound of Sauer, Perles and Shelah and Vapnik and Chervonenkis, equiangular lines, Fourier coefficients, Wronksian, Petersen graph)

Welcome. This is an Honours level course that both covers the material of MATH 221 and MATH 152 but goes at a faster pace and 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.

Our Section 101 meets 10:00-10:50 in Math Annex 1100. My office is nearby, Math Annex 1114.
I will be available for office hours MWF 11-12 and 12-1 after class and often at 1 as well. I and also have booked MATH 102 for special office hour/tutorial 5-6 Thursdays before assignments are due. Thus the first session will be Sept. 16. While I will be available for questions, I hope you begin talking to each other in that venue (but keep your masks on!).

I hope that the class will be almost entirely in person but Covid issues may intervene. Masks will be mandatory in classrooms. As a student, you won`t be able to eat in the classroom but can take brief sips of water if needed. I do hope students will continue to answer and ask questions in class but they will have to speak up more than usual. While vaccinations are not currently mandated, you will be restricted from number of activities without full vaccination. Planes/Trains for example or indoor restaurants (it does rain in Vancouver).
The room Math Annex 1100 has decent ventilation (they have tested this) and there are windows that can open to provide additional ventilation. I expect the midterms to be in person. We will have more information as the term begins.
For those unable to attend a lecture in person (surely you don`t wish to miss a lecture! But for example if some symptoms require you to self isolate. . . ), I typically post decent lecture notes after the fact. Attending in person is valuable for you.

Midterm 1 is scheduled for Friday October 8 (before the thanksgiving holiday) and Midterm 2 is scheduled for Wednesday Nov 17 after the fall break (Wednesday Nov 10 through Friday Nov 12).

  • Course Outline  

  • Assignment 1 due September 17 by class time. Solutions  
  • Assignment 2 due September 24 by class time. Solutions  
  • Assignment 3 due October 1 by class time. Solutions  
  • Assignment 4 due October 15 by class time. Solutions  
  • Assignment 5 due October 22 by class time. Solutions  
  • Assignment 6 due October 29 by class time. Solutions  
  • Assignment 7 due November 5 by class time. Solutions  
  • Assignment 8 due November 22 by class time. Solutions  
  • Assignment 9 due Dec 1 by class time. This will be the last assignment. Solutions  
  • Sample Midterm 1. Our first midterm is an in class midterm October 8. We should be able to give you 55 minutes since there is no class immediately afterwards. I will not post solutions but you can ask me about your solutions during office hours.  
  • Midterm 1 Solutions. There were a number of errors in computing various things. The Gaussian Elimination question was done well with minor errors. The eigenvalue problem had a number of students unable to compute the characteristic polynomial det(A-\lambda I). In question 3, it would have been nice if people had computed A^n=MD^nM^{-1} even if unsure how to use it. The limit problem was not well liked!  
  • Sample Midterm 2. Please note that question 2 is about diagonalizing a 2x2 matrix but with Complex numbers. I would have preferred a replacement for question 2 where you are given the diagonalization and use it to solve a system of two differential equations. Again this will be an in class test and I have requested to borrow a few extra minutes (5?) from the following class.  
  • Midterm 2 Solutions.  
  • 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.  

  • Lecture 1 Sept 8 Introduction to matrices and matrix products  
  • Lecture 2 Sept 10 Multiplicative inverses  
  • Putnam Problem. Sept 10 You are not expected to solve such problems on tests. There are three ideas (!) that enable the problem to be solved. Our assignments/tests will have some theoretical problems that will often require some cleverness.  
  • Lecture 3 Sept 13 Linear transformations  
  • Lecture 4 Sept 15 The basic ideas for eigenvalues/eigenvectors.  
  • Lecture 4 Sept 15 Bird example (Leslie Matrix) and eigenvalues/eigenvectors.  
  • Lecture 5 Sept 17 eigenvalues/eigenvectors and computing Bird populations (Leslie Matrix).  
  • Lecture 5/6 Sept 17 White and Blue coordinates (change of coordinate system)  
  • Lecture 6 Sept 20 Fibonacci numbers  
  • Lecture 6 Sept 20 Repeated Averaging  
  • Lecture 7,8 Sept 22,24 Gaussian Elimination, an organized approach to solving systems of equations.  
  • Lecture 9 Sept 27 Computing a matrix inverse.  
  • Lecture 10 Sept 29 Definition of a Determinant  
  • Lecture 11/12 Oct 1,4 Proofs about determinant function.  
  • Lecture 13 Oct 6 Proof that partial fractions work using Vandermonde determinant.  
  • Lecture 13 Oct 6 Proof that interchanging first two rows changes sign of determinant.  
  • Lecture 14 Oct 13 Axioms for Fields and Vector Spaces  
  • Lecture 14 Oct 13 Vector Spaces. A vector is not a vector?  
  • Lecture 15 Oct 15 Linear Independence  
  • Lecture 16 Oct 18 Basis and Dimension of Vector Spaces.  
  • Lecture 17 Oct 20 A proof of a bound using the Dimension of a Vector Spaces. A `trip to the opera`, not examinable.  
  • `A vector may have a life beyond its role as a vector in a vector space.` We have given you examples of vectors, in particular functions, matrices etc, which have additional interpretations beyond what comes from considering them as vectors in a vector space. The value of considering them as vectors in a vector space lies in the ability to use our vector space ideas including dimension.
  • Lecture 18 Oct 22 Basis and Dimension and the Nullity theorem  
  • Lecture 19 Oct 25 Coordinates. Vectors spaces of dimension k over R are `thinly veiled` examples of R^k.  
  • Lecture 20 Oct 27 Change of basis including similarity ideas.  
  • Lecture 20 Oct 27 How to extend a set of linearly independent vectors to a vector space.  
  • Lecture 21 Oct 29 A change of basis matrix is invertible.  
  • Lecture 21 Oct 29 System of Differential Equations  
  • Lecture 22 Nov 1 Introduction to Complex Numbers  
  • Lecture 23 Nov 3 More Complex Numbers and Differential Equations  
  • Lecture 23 Nov 3 System of Differential Equations  
  • Lecture 24 Nov 5 Basics of vector geometry.  
  • Lecture 25 Nov 8 Inner product spaces.  
  • Lecture 25 Nov 8 Orthogonal vector spaces. Inner product spaces.  
  • Lecture 26 Nov 15 Equiangular Lines. A `trip to the opera`, not examinable.  
  • Lecture 27 Nov 19 Gram Schmidt and orthonormal bases.  
  • Lecture 28,29 Nov 22,24 Orthogonal Projections and Least Squares. Also Fourier Coefficients.  
  • Lecture 30 Nov 26 Orthogonal diagonalization of symmetric matrices.  
  • Lecture 31 Nov 29 Unitary diagonalization of Hermitian matrices. Gram Schmidt applied to complex vectors will not be on final.  
  • Lecture 32 Dec 1 Quadratic Forms.  
  • Lecture 33 Dec 3 Complex numbers go in circles. We indicate why our point averaging problem from Lecture 6 tends to an ellipse. This is mostly a `trip to the opera` and the details not examinable but of course many of the ideas have already appeared in other contexts and are examinable (dominant eigenvalues/dominant eigenvectors, complex conjugates for DE`s etc).  
  • Lecture 34 Dec 6 Back of Sphere Front of Sphere Whole Sphere  

  • Other Materials (in a kind of reverse order, recently added material first)

  • Moore Graphs special graphs (which include Petersen) with special eigenvalue properties that force them to be rare.  
  • Digraphs Some basic properties of digraphs and their adjacency matrices.  
  • The Wronskian using derivatives to determine linear independence of functions.  
  • Brief discussion of Jordan Blocks. All matrices can be `diagonalized` into Jordan Blocks, namely the diagonal matrices are Jordan Blocks. This is called Jordan Canonical Form since there is only one `diagonal` form apart from simultaneous row and column permutations. Not examinable.  
  • Proof of Fischer`s block design inequality using rank. This is the only known proof. Not examinable.  
  • Examples of proofs by Mathematical Induction  
  • There are three 2x2 matrices Not examinable.  
  • Levels of Generality in this course: thinking of linear transformations, manipulating matrices and dealing with the entries of a matrix  
  • Matrix multiplication and some interpretations.  
  • Mathematical humour jokes from Chris Ryan, now a professor in Operations and Logistics, Sauder School of Business, UBC.  
  • A surprisingly nice problem concerning divisibility.