625.609.81 - Matrix Theory

Applied and Computational Mathematics

Spring 2024

Description

This course focuses on the fundamental theoretical properties of matrices. Topics will include a rigorous treatment of vector spaces (linear independence, basis, dimension, and linear transformations), orthogonality (inner products, projections, and Gram-Schmidt process), determinants, eigenvalues and eigenvectors (diagonal form. of a matrix, similarity transformations, and matrix exponential), singular value decomposition, and the pseudo-inverse. Essential proof writing techniques and logic will be reviewed and then used throughout the course in exams and written assignments. Prerequisite(s): Multivariate calculus

Course Structure

The course materials are divided into 14 modules which can be accessed on the course Home page or by clicking Modules on the course menu. Each module has several items including lectures, readings, group activity and/or discussion, and assignments. Modules begin Mondays at 12:00 am Eastern Time and run for a period of seven (7) days. Please check the Calendar and Announcements regularly for specific assignment due dates and other important information.

Course Topics

Basic Matrix Operations and Mathematical Proofs

Vector Spaces, Subspaces, Linear Independence, Span, Basis, and Dimension

Linearity, Rank, Nullity, and Isomorphisms

Matrices of Linear Transformations and Change of Basis

Elementary Operations, Row (Reduced) Echelon Form, and LU-Factorization

Trace, Determinant, and Solving Linear Systems

Inner Product Spaces and Norms

Orthogonality, QR-Factorization, and Least Squares

Eigenvalues, Eigenvectors, and Diagonalization

Schur Triangularization and Jordan Normal Form.

Adjoint, Unitary Operators, and Normal Operators

Singular Value Decomposition and Pseudoinverse

Course Goals

To understand the fundamental theorems, tools, and techniques in Linear Algebra and Matrix Theory and to hone logical and mathematical thinking.

Course Learning Outcomes (CLOs)

Compose rigorous arguments to prove or disprove mathematical statements.

Determine the number of solutions to a system of linear equations and solve the system.

Examine matrices for key properties (e.g., rank, invertibility, associated eigenspaces, diagonalizability, normality, etc.).

Apply algorithms to transform. matrices or sets of vectors to more computationally useful forms (e.g., Gaussian elimination, Gram-Schmidt algorithm, etc.).

Compare the advantages of and uses for various matrix factorizations (e.g., LU, QR, Schur triangularization, Jordan Normal Form, SVD, etc.).

Explain the relationship between matrices, linear transformations, similarity, and change of basis.

Analyze sets of polynomials and functions as vector spaces with inner products and norms.

Textbooks

Nair, M. T., Singh, A. (2019). Linear Algebra. Singapore: Springer Singapore.

eBook ISBN: 978-981-13-0926-7

Hardcover ISBN: 978-981-13-0925-0

Softcover ISBN: 978-981-13-4533-3

Note: A digital copy of this book is available with your JHU login through the library website (see further information below).

Student Coursework Requirements

It is expected that each module will take approximately 10–15 hours per week to complete. An approximate breakdown of required activities: reading the assigned sections of the texts as well as other supplementary readings (approximately 2-4 hours per week), watching the lecture videos and working through the in-lecture examples and exercises (approximately 2-4 hours per week), group activities (approximately 1 hour per week), and problem set assignments (approximately 5–6 hours per week). Other recommended activities include: reviewing feedback on previous assignments, working through practice problems, emailing and/or meeting with the professor as needed. This course has the following basic student requirements:

Assignments (30% of Final Grade Calculation)

Assignments consist of a problem set that students must solve and write up. Assignments should be done neatly on paper and scanned or completed electronically using LaTeX or a tablet device. Your submission should be uploaded to Gradescope, using the built in tool to assign pages to problems. This gives you a chance to ensure there are no pages missing from your assignment, and it makes it easier for the grader to find your work. Failure to appropriately assign pages to problems may result in a deduction of points. For further information on how to submit an assignment to Gradescope, please visit the Submitting a PDF Gradescope help article.

Collaborations and discussions between students are key ingredients to success in a graduate course. The assignments will be challenging and thought-provoking, and I encourage you to work together on them. However, while you are welcome to discuss how to approach and solve the assigned problems, you are expected to submit a solution that you have written up on your own that reflects your understanding of the problem. If you work with someone, you must make a note at the top of the first page of your assignment (e.g., "Collaborated with Jane Doe").

In addition to collaborating with your peers, you may use the assigned readings and any content on the Canvas site. You may reference other textbooks or general Matrix Theory resources, but you may not search for solutions to assigned problems (for further details, please see the Course Policies section of the Syllabus). Unless explicitly stated otherwise, you should not use computational assistance (e.g., programming languages) to solve assigned problems (beyond checking basic arithmetic). If you have questions or are stuck on an assignment, please reach out to me and I will be happy to help!

Eff ectively communicating mathematics and proof-writing are a significant focus of this course. In addition to logical and mathematical correctness, there will be an emphasis on clarity of solutions and proper use of notation and vocabulary in the grading.

All assigned problems can be solved using only information from the readings, lectures, and other materials on Canvas up to that point in the course. You should not use results, concepts, or techniques we have not yet covered in your solutions.

Assignments are due according to the dates posted on the Canvas course site. You may check these due dates in the Course Calendar or under the Assignments in the corresponding modules. We will aim to have assignments graded and feedback posted about one week after assignment due dates. Grades for assignments can be viewed on both Canvas and Gradescope, but detailed feedback will be posted exclusively through Gradescope; for further information, please visit the Viewing your Submission Gradescope help article.

Unless you have a previously arranged extension or there are exceptional extenuating circumstances, late submissions will not be accepted and will receive a grade of zero. For further detail on extensions, please see the Course Policies section of the Syllabus.

Group Activities (15% of Final Grade Calculation)

Group activities consist of one or two problems that highlight foundational concepts from the module. They serve as a good self-check to ensure you are understanding some the basic techniques before tackling the module's assignment.

Solutions to group activity problems must be posted as a pdf attachment to your assigned group discussion board on Canvas by Friday at 11:59 pm Eastern Time. Two reply posts, comparing your solutions to your group mates', must be posted by Sunday at 11:59 pm Eastern Time. Reply posts should follow the 3C+Q Method outlined below:

3C+Q Method

Compliment - Praise a specific aspect of the post. For example: "I like that your solution..."

Connect - Build connections to other relevant course materials (outside the current module if possible). For example: "This seems to relate to X from the lecture/reading/Module Y in that..."

Comment - Add a statement of agreement or disagreement. For example: "What I would add to your post is that..." or "I might come to a diff erent conclusion because..."

Question - Keep the conversation going by asking a specific question about the topic under discussion. For example: "What eff ect might X have on..." or "Would this work if..."

Solutions to group activity problems are graded based on eff ort and completeness more than correctness. A complete solution must be written up neatly and include all relevant justification for how you arrived at your answer (the same as for assignments). Reply posts are graded based on following the 3C+Q method, engagement with peers, and furthering the discussion. You may receive partial credit on late group activity posts. For information on how to view the full rubric, please visit the Viewing the Rubric for a Graded Discussion Canvas help page.

Grades and feedback for Group Activities will be through Canvas. For information on how to view feedback on these, please visit the How do I view assignment comments from my instructor? Canvas help page.

I encourage you to use the group activity boards beyond the required posts as a springboard to discuss the course content and collaborate with your peers on assignments, but this is not necessary in order to receive full credit (as long as the posts you do make are of good quality).

Applications Discussions (5% of Final Grade Calculation; 2.5% each)

There are two applications discussions: the first taking place in Modules 5-6 and the second taking place in Modules 12-13. Each of these requires students to pick a specific application of matrix theory that interests them, research it, compose a 2-3 paragraph summary, and post their write-up to the appropriate discussion board on Canvas by Friday at 11:59 pm Eastern Time of the second module (Module 6 and Module 13 respectively). Two reply posts engaging with other students who researched diff erent applications must be posted by Sunday at 11:59 pm Eastern Time. Reply posts should follow the 3C+Q method as outlined below:

3C+Q Method

Compliment - Praise a specific aspect of the post. For example: "I like that your write-up..."

Connect - Build connections to relevant course materials. For example: "This seems to relate to X from the lecture/reading/Module Y..."

Comment - Add a statement of agreement or disagreement or expand on something the post mentioned. For example: "I notice from source X you referenced that..." or "I found another source Y that..."

Question - Keep the conversation going by asking a specific question about their application. For example: "How does this relate to..." or "Could this also be used for..."

Exams (40% of Final Grade Calculation; 20% from the Midterm Exam, 20% from the Final Exam)

There are two exams: a Midterm in Module 7 and a cumulative Final in Module 14. They are released as soon as the module they are in becomes available (i.e., midnight Eastern time on Monday) and are due at the typical assignment deadline at the end of the module (i.e., 11:59 pm Eastern time Sunday). You may spend as much time as you would like on the exam within that time window. Exams are the only item in their modules; there are no other readings, videos, or assignments. For exams, you may use the textbook as well as anything on the Canvas site. You may not consult anyone or anything else (beyond asking the instructor for clarification if needed). Exams must represent an individual eff ort by you alone.

In order to unlock each exam, you will be required to read the full rules and agree to an honor statement that you will abide by the rules.

Exam submissions should be uploaded to Gradescope, using the built in tool to assign pages to problems. Failure to appropriately assign pages to problems may result in a deduction of points. For further information on how to submit an assignment to Gradescope, please visit the Submitting a PDF Gradescope help article. Detailed feedback on exams will also be posted through Gradescope; for further information, please visit the Viewing your Submission Gradescope help article.

Grades for exams can be viewed on both Canvas and Gradescope, but detailed feedback will be posted exclusively through Gradescope; for further information, please visit the Viewing your Submission Gradescope help article.

Late exams will not be accepted, nor will extensions on exams be granted unless there are exceptional circumstances.

Grading Policy

Assignments are due according to the dates posted on the Canvas course site. You may check these due dates in the Course Calendar or the Assignments in the corresponding modules. Grades and feedback will typically be posted about one week after assignments are due.

Eff ectively communicating mathematics through proof-writing is a significant focus of this course. In addition to logical and mathematical correctness, there will be an emphasis on clarity of solutions and proper use of notation and vocabulary in the grading of assignments.

A grade of A indicates achievement of consistent excellence and distinction throughout the course—that is, conspicuous excellence in all aspects of assignments and discussion in every week. A grade of B indicates work that meets all course requirements on a level appropriate for graduate academic work.

Course grades will be based upon accumulated points as follows:

Score RangeLetter Grade

100-98     = A+

97-94       = A

93-90       = A-

89-87       = B+

86-83       = B

82-80       = B-

79-77       = C+

76-73       = C

72-70       = C-

69-67       = D+

66-63       = D

< 63 = F

Overall course grades will be determined by the following weighting:

Coursework Category                           Percentage of Course Grade

Group Activities                                             15%

Problem Set Assignments                               30%

Application Discussions                                   5%

Midterm Exam                                               20%

Final Exam                                                    20%

Highest grade of the above                             10%

Note that 10% of your grade is based on your highest grade across these categories. This is meant to off er flexibility and emphasize each student's unique strengths. For example, if your grades are A in Group Activities, B+ on Homework Assignments, A- on the Application Discussions, B on the Midterm Exam, and B+ on the Final Exam, Group Activities would contribute 25% rather than the usual 15% to your overall grade computation.