Math 2R03 - Theory of Linear Algebra
(Fall 2024)

MATH 2R03 is a second course in linear algebra. From the calendar:

In this course we will focus on the theoretical underpinnings of linear algebra, and in particular, we will emphasize the proofs of linear algebra. The main objectives of this course are: The prerequisite for this course is one of MATH 1AA3, 1LT3, 1NN3, 1XX3, 1ZB3, ARTSSCI 1D06 A/B, ISCI 1A24 A/B; and one of MATH 1B03, 1ZC3, 1ZZ5.


News

Schedule

Homework

Handouts

Grading Scheme

Policies

Course Information

Important Note:

The purpose of this webpage is to collect together all of the needed resources for the course. You should also consult the Avenue-To-Learn page and Mosaic where you will find room information.

Instructor:

Adam Van Tuyl

Office: Hamilton Hall 419 NOTE: I'm currently using HH 326
Office Hours: Wednesdays 9:30-10:20
Email: vantuyl@math.mcmaster.ca

TA:

Silas Vriend
Email: vriendsp@mcmaster.ca
Office Hours: Thursdays 12:30-1:30; Fridays 12:00-1:00 in Math Cafe

Course Outline:

The official course outline is found on SimpleSyllabus

Place and Time:

This course will be offered in-person. The times are:

Lecture C01: Monday, Wednesday 8:30-9:20, Friday 10:30-11:20
Tutorial T01 : Monday 10:30-11:20
Tutorial T02 : Tuesday 1:30-2:20

For locations of the class and tutorials, see Mosaic

Textbook:

Linear Algebra Done Right (Fourth Edition)
by Sheldon Axler

This book can be download for free.

Here is a link to the book on the campus bookstore website: Purchase at Campus Bookstore

Important Links:

The following online resources will be used during this course:



News (Last Updated: Nov 20, 2024)

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Schedule

We will be using the following schedule.

Note 1: There may be changes so please refer back to this page to stay up-to-date

Note 2: The videos are provided as a resource -- they are based upon the third edition of the textbook (we are using the fourth edition). Our lectures may cover different topics. A detailed schedule is given below.


Math 2R03 Schedule
Week 1 (Sept 2-6)
Homework: Assignment 0 (Due Sept 6)
Lecture Topic Video Suggested Problems
Lecture 1 1.A R^n and and C^n Video 1.A - 10, 11, 15
Lecture 2 1.B Vector Spaces Video 1.B - 1, 2, 3
Week 2 (Sep 9-13)
Homework: Assignment 1 (Due Sep 13)
Lecture 3 1.C Subspaces I Video 1.C - 6, 10, 23
Lecture 4 1.C Subspaces II
Writing Proofs
Video 1.C - 10, 19, 20
Lecture 5 2.A Span and Linear
Independence I
Video 2.A - 2, 3, 7
Week 3 (Sep 16-20)
Homework: Assignment 2 (Due Sep 20)
Lecture 6 2.A Span and Linear Independence II Video 2.A - 3, 13
Lecture 7 2.B Bases Video 2.B - 3, 7, 10
Lecture 8 2.C Dimension Video 2.C - 11, 14, 17
Week 4 (Sep 23-27)
Homework: Assignment 3 (Due Sep 27)
Lecture 9 3.A Vector Space of Linear Maps Video 3.A - 2, 4, 7
Lecture 10 3.B Null Spaces and Ranges I Video 3.B - 2, 6, 9
Lecture 11 3.B Null Spaces and Ranges II
3.C Matrices
Video 3.C -2, 5, 6
Week 5 (Sep 30-Oct 4)
Homework: Study for Midterm 1 (Oct 4)
No class Sep 30
Lecture 12 3.D Invertibility and Isomorphic Vector
Spaces I
Video 3.D - 2, 3, 5
Lecture 13 Midterm 1 (Oc. 4)
Week 6 (Oc 7-11)
Homework: Assignment 4 (Due Oct 15)
Lecture 14 3.D Invertibility and Isomorphic Vector
Spaces II
Video 3.D - 11, 14,
15 (Hint: why is T invertible?)
Lecture 15 4 Polynomials Video 4 - 4, 5, 6
Lecture 16 5.A Invariant Subspaces I Video 5.A - 7, 8, 9
Week 7 (Oct 14-18)
Reading Week (No Classes)
Week 8 (Oct 21-25)
Homework: Assignment 5 (Due Oct 25)
Lecture 17 5.A Polynomials Applied to Operators
5.B Minimal Polynomials
Video 5.A - 2, 5, 21, 23
Lecture 18 5.B Minimal Polynomials No Video 5.B - 1, 3, 4
Lecture 19
5.B Minimal Polynomials II
No Video 5.B - 6, 10, 11
Week 9 (Oct 28-Nov 1)
Homework: Assignment 6 (Due Nov 1)
Lecture 20 5.C Upper Triangular Matrices Video
Based on 5.B
in 3rd ed
5.C - 1, 3, 6, 8
Lecture 21 5.D Diagonalizable Opeators Video
Based on 5.C
in 3rd ed
5.D - 2, 3, 5, 7
Lecture 22 8.A Generalized Eigenvectors and
Nilpotent Operators I
Video 8.A - 4, 6, 7, 8
Week 10 (Nov 4 -Nov 8)
Homework: Assignment 7 (Due Nov 8)
Lecture 23 8.A Generalized Eigenvectors and
Nilpotent Operators II
Video 8.A - 12, 13, 17, 21
Lecture 24 8.B Generalized Eigenspace Decomposition Video 8.B - 2, 4, 5
Lecture 25 8.B Generalized Eigenspace Decomposition II Video 8.B - 7, 9, 11, 12
Other Videos based upon third edition
may also be useful, including material on the
minimal and characteristic polynomial
Video
Video
Video
Week 11 (Nov 11-15)
Homework: Study for Midterm 2 (Nov 15)
Lecture 26 6.A Inner Products and Norms I Video 6.A - 3, 4, 5, 9
Lecture 27 6.A Inner Products and Norms II Video 6.A - 13, 17, 21, 26
Lecture 28 Midterm II (Nov 15)
Week 12 (Nov 18-22)
Homework: Assignment 8 (Nov 22)
Lecture 29
6.B Orthonormal Bases Video 6.B - 2a, 3, 7, 8a
Lecture 30 6.B Linear Functionals Video 6.B - 11, 12
Lecture 31 6.C Orthogonal Complements
and Minimization
No Video 6.C - 1, 2, 3, 5
Week 13 (Nov 25-29)
Homework: nothing due this week, next assingmet due next Monday
Lecture 32 7.A Self-Adjoint and Normal Operators I Video 7.A - 1, 2, 3
Lecture 33 7.A Self-Adjoint and Normal Operators II Video 7.A - 6, 7, 15
Lecture 34 7.B The Spectral Theorem Video 7.B - 1, 2, 10, 14
Week 14 (Dec 2-4)
Homework: Assignment 9 (Due Dec 2)
Lecture 35 5.D - Gershgorin Disk Theorem No video
Lecture 36 Buffer/Review No video

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Homework

There will be nine homework assignments (your lowest two marks will be dropped). Assignments are posted below. Assignments will be submitted via Crowdmark. You will receive an email to your McMaster account that you will use to upload your assignment. All assignments due by 11:59PM on the due date.

For more information on writing proofs, the following notes may help:

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Handouts

All class handouts are available as PDF files (except where indicated)

Course Information
Course handout describing the course (available online)

Midterm 1 Review Sheet
Handout describing first midterm.

Midterm 2 Review Sheet
Handout describing second midterm.

Exam Review Sheet
Handout describing final exam.

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Grading Scheme

I will calculate your mark using two different weightings. Your final mark will be the higher of the two weights.


Weighting 1
20% = 9 Assignments (best 7 used)
40% = Midterms (2 x 20%)
40% = Final Exam

Weighting 2
20% = 9 Assignments (best 7 used)
20% = Maximum of Midterm 1 and 2
60% = Final Exam

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McMaster Polices

There are number of important McMaster Policies that you should be aware of. For a complete and up-to-date list of polices, see the official Course Outline.


With respect to MSAFs, missed work will be moved to final, as described in the course outline.


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