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MATH 1B03 – Linear Algebra I

Course Description

● From the academic calendar (2021-22):

Vector spaces given by solutions to linear systems. Linear independence, dimension. Determinants.

Eigenvalues, eigenvectors and diagonalisation. Complex numbers.

● Three lectures, one tutorial; one term

Prerequisite(s): Grade 12 Calculus and Vectors U or MATH 1F03

This course is an introduction to linear algebra. We are interested in both a computational approach (e.g., computing solutions to a linear system of equations) and a theoretical approach (e.g., an understanding of the underlying idea of a vector space). For the Fall 2021 academic term, the course will be team taught with material primarily presented through prerecorded lectures, initially by Dr. Sawyer. Class and tutorial times will be devoted to problem solving, quizzes, and two synchronous lectures each week for each section by Dr. Le and Dr. Sroka, that repeat some of the recorded lectures in a different way, and provide for interaction with students. The three instructors may switch roles throughout the term. There will be weekly homework, and computer labs for self-assessment, and Teaching Assistants to help with these. There will be biweekly assignments, optional multiple choice quizzes, a Midterm Test and a Final Exam for the purpose of determining your final grade in the course.

Course and Learning Objectives

Course Objectives

MATH 1B03 is the first course on linear algebra. By the end of this course, students should be able to:

● do computations involving matrices. For example, you should be able to solve systems of linear equations using Gauss-Jordan elimination, to be comfortable with matrix arithmetic, to compute determinants, and to find eigenvalues/eigenvectors of a matrix. Homework and labs will facilitate this objective, as well as assignments and quizzes.

● explain some theoretical underpinnings of linear algebra. For example, you should be able to understand the language of vector spaces to develop a theory that supports and describes what is observed in the computations above. As well, you will practice critical thinking skills by demonstrating understanding of the concepts encountered in both computational and theoretical contexts. Homework, labs, and assignments will facilitate this objective.

Materials & Fees

Required Materials/ Resources

Textbook Information:

● (Required) We will be using Linear Algebra and its Applications (6th Edition) by D. Lay, S. Lay, and J. McDonald, as well as the publishers’ MyLab Math, so it is required that you purchase access to this feature and the textbook combined, which can be done at the campus bookstore. You can also purchase the loose-leaf version through their website after purchasing access to MyLab Math as described above.

● (Optional) Student Solutions Manual for Elementary Linear Algebra - Applications Version.

Virtual Course Delivery

To follow and participate in virtual classes it is expected that you have reliable access to the following:

● A computer that meets performance requirements found here.

● An internet connection that is fast enough to stream video.

● Computer accessories that enable class participation, such as a microphone, speakers and webcam when needed.

If you think that you will not be able to meet these requirements, please contact [email protected] as soon as you can. Please visit the Technology Resources for Students page for detailed requirements. If you use assistive technology or believe that our platforms might be a barrier to participating, please contact Student Accessibility Services, [email protected], for support.

Course Overview and Assessment

Topics

We will cover the following topics: vector spaces given by solutions to linear systems; linear independence; dimension; determinants; eigenvalues and eigenvectors; diagonalisation; and complex numbers.

Course Delivery:

The course will be delivered using both asynchronous and synchronous components. The asynchronous component consists of video lectures of the course material (posted on Avenue to Learn). For the synchronous component, we will use the scheduled class time as follows:

● problem solving sessions based upon the online lectures of the previous week.

● instructors will be online some of the time to redeliver some of the lecture content in another way, to answer questions, and the remaining time can be used by you to watch the video lectures and review the pdf files of whiteboard pictures taken during the recorded lectures.

MATH 1B03 (Provisional) Calendar – Fall 2021

We will be using the following schedule. Please note that there may be changes; always refer to Avenue-to-Learn for the latest information.

Week	Lecture	Topics	Key Deadlines
1 - (Sept 7-10)	Lecture 1	Introduction   1.1 Systems of Linear Equations
	Lecture 2	1.2 Row Reduction and Echelon Forms
2 - (Sept 13-17)	Lecture 3	1.2 Row Reduction and Echelon Forms (Continued)   Introduction to MyMathlab/Octave	ASSIGNMENT #1: Due at 11:59pm on   September 19
	Lecture 4	1.3 Vector Equations
	Lecture 5	1.4 Matrix Equation Ax = b
3 - (Sept 20-24)	Lecture 6	1.5 Solution Sets of Linear Equations	OPTIONAL MULTIPLE CHOICE QUIZ   #1 during your first lecture slot of   the week
	Lecture 7	1.7 Linear Independence
	Lecture 8	1.8 Introduction to Linear Transformations
4 - (Sept 27-Oct 1)	Lecture 9	1.9 Matrix of a Linear Transformation	ASSIGNMENT #2: Due at 11:59 on   October 3
	Lecture 10	1.6 Applications of Linear Systems
	Lecture 11	2.1 Matrix Operations
5 - (Oct 4-8)	Lecture 12	2.2 The Inverse of a Matrix	OPTIONAL MULTIPLE CHOICE QUIZ   #2 during your first lecture slot of   the week
	Lecture 13	2.2 The Inverse of a Matrix (continued)   2.3 Characterizations of Invertible Matrices
	Lecture 14	2.3 Characterizations of Invertible Matrices (continued)   2.4 Partitioned Matrices
6 - (Oct 11-17)                                                                   Midterm Recess – no classes
7 - (Oct 18-22)	Lecture 15	2.7 Applications to Computer Graphics	ASSIGNMENT #3: Due at 11:59pm on   October 24
	Lecture 16	3.1 Introduction of Determinants
	Lecture 17	3.2 Properties of Determinants
8 - (Oct 25-29)	Lecture 18	3.3 Cramer's Rule, Volume, and Linear   Transformations	MIDTERM TEST: during the first   lecture of week in class time
	Lecture 19	4.1 Vector Spaces and Subspaces
	Lecture 20	4.1 Vector Spaces and Subspaces (continued)   4.2 Null Spaces, Column Spaces, and Linear Transformations
9 - (Nov 1-5)	Lecture 21	4.2 Null Spaces, Column Spaces, and Linear Transformations (continued)	ASSIGNMENT #4: Due at 11:59 pm   on November 7
	Lecture 22	4.3 Linear Independent Sets and Bases
	Lecture 23	4.4 Coordinate Systems
10 - (Nov 8-12)	Lecture 24	6.1 Inner Product, Length, and Orthogonality   6.2 Orthogonal Sets	OPTIONAL MULTIPLE CHOICE QUIZ   #3 during your first lecture slot of   the week
	Lecture 25	6.3 Orthogonal Projections   6.4 Gram-Schmidt Process
	Lecture 26	4.5 Dimension of a Vector Space
11 - (Nov 15-19)	Lecture 27	4.5 Dimension of a Vector Space (continued) (Section 4.6 in 5th Edition)	ASSIGNMENT #5: Due at 11:59pm on   November 21
	Lecture 28	5.1 Eigenvectors and Eigenvalues
	Lecture 29	5.2 The Characteristic Equation
12 - (Nov 22-26)	Lecture 30	5.3 Diagonalization	OPTIONAL MULTIPLE CHOICE QUIZ   #4 during your first lecture slot of   the week
	Lecture 31	5.3 Diagonalization (Continued)
	Lecture 32	5.4 Eigenvectors and Linear Transformations
13 - (Nov 29-Dec 3)	Lecture 33	Appendix B Introduction to Complex Numbers	ASSIGNMENT #6: Due at 11:59pm on   December 5
	Lecture 34	5.5 Complex Eigenvalues
	Lecture 35	5.6 Discrete Dynamical Systems
14 - (Dec 6-8)	Lecture 36	5.9 Applications to Markov Chains (Section 4.9 in 5th Edition)
	Lecture 37	Review

Self assessment

● Homework may be handed in for immediate feedback.

● Computer labs will also be available in selected weeks, and a schedule will be released on Avenue to Learn at a later date. The labs will use either MatLab or Octave, which are available free of charge. They will not be graded for assessment purposes, but teaching assistants will be available to check students’ work and help students learn how to use these programs as an alternative to calculations by hand.

Evaluation

Assignment Information:

There will be six assignments made available through online submission to Crowdmark. A link to the assignments will be on Avenue-to-Learn. Each assignment will consist of two to four questions requiring written answers, and just one of the questions will be marked. See the calendar above for due dates.

Quizzes:

There will be four Optional Multiple Choice Quizzes conducted on MyLab Math. Quizzes will be done during your scheduled class time (you must write with your own section) on MyLab Math. At the end of the quiz, the student may choose to submit the quiz for marking, and in this case the quiz grade will count 5% toward the final grade in the course. Or instead, at the end of the quiz, the student may choose to not submit the quiz for marking, and in this case the evaluation weight of the quiz will be added to that of the final exam.

Midterm Test Information:

There will be both a multiple choice component conducted on MyLab and a take-home component submitted through Crowdmark in the last week of October. The takehome component will consist of two to four questions, all of which will be marked for assessment.

Final Exam Information:

The final examination will consist of a 1 hour multiple choice component scheduled by the registrar, and a takehome component consisting of four to six questions to be done within 48 hours, and all questions will be marked for assessment. The registrar will publish more information on the exams at a later date. The exam will cover all the material from the course; details on topics covered will be announced on Avenue.

Marking Scheme Information.

Your final mark will be calculated as follows:

Assessment	Weight	Notes
1. Final Examination	35%	Scheduled by registrar
2. Midterm Test	30%	During class time
2. Optional Quizzes	20%	4 at 5% each
3. Assignments	15%	Best 5 at 3% each

The weights for any of the optional quizzes that you decide not to hand in will be reassigned to the final exam.

Course Support:

In order to help you succeed in this course, the following services are available to you.

● Practice Problems. Suggested homework problems and practice tests/exams will be made available on Avenue / MyLab Math.

● Tutorials. There are six one hour tutorials each week, and you are encouraged to attend at least the two scheduled for your section. The tutorials are intended to provide additional material to help students learn the course material, and provide opportunities to ask additional questions and seek help. Although attendance in tutorials is not mandatory, it is strongly encouraged. Tutorial information to be announced.

● Drop-In Centre. More personalized assistance can be obtained by coming to the Math Drop-In Centre on the first floor of Hamilton Hall. It is expected that an online form of the Drop-In Centre will be available in 2021-22. Tutors are freely available to assist with linear algebra questions. More detailed times and information is available on their web site: https://www.math.mcmaster.ca/undergraduate/math-drop-in-centre.html