625.609.81 - Matrix Theory

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!

Effectively 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 different conclusion because..."
  • Question - Keep the conversation going by asking a specific question about the topic under discussion. For example: "What effect might X have on..." or "Would this work if..."

Solutions to group activity problems are graded based on effort 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 different 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 effort 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.

Effectively 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 Range	Letter 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 offer 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.

Course Policies

While the lecture videos, readings, and other provided materials contain all the information you need to solve any assigned problems, you are allowed to consult other references (e.g., other textbooks) to strengthen your understanding of general concepts if you wish (except during exam weeks). Taking or modifying solutions from any source is not permitted. This includes but is not limited to solutions manuals (to any text), websites posting problem solutions (e.g., Chegg, Course Hero, Stack Exchange, etc.), other people, and/or AI resources. Not only does this violate the Academic Integrity policy, it hurts your own learning and understanding. If you need assistance, I am more than happy to provide it!

In the case that you are consulting another textbook for general information and accidentally find a solution to an assigned problem in such a reference, DO NOT read it and work out the solution on your own.

Each student is granted four 24-hour extensions on assignments over the course of the semester, no explanation needed. You may use up to two 24-hour extensions on the same assignment (granting a 48-hour extension). If you wish to use an extension, please notify me (the instructor) via email so that I can add the extension to Gradescope. If you do not notify me in time for me to add the extension to Gradescope, it is your responsibility to email me your submission by the extended deadline. Extensions may be applied only to problem set assignments, not to group activities, application discussions, or exams.

If you need extensions beyond the default four 24-hour extensions, you must coordinate them with me (the instructor) BEFORE the assignment is due. Approval of such extensions is not guaranteed.

Academic Policies

Deadlines for Adding, Dropping and Withdrawing from Courses

Students may add a course up to one week after the start of the term for that particular course. Students may drop courses according to the drop deadlines outlined in the EP academic calendar (https://ep.jhu.edu/student-services/academic-calendar/). Between the 6th week of the class and prior to the final withdrawal deadline, a student may withdraw from a course with a W on their academic record. A record of the course will remain on the academic record with a W appearing in the grade column to indicate that the student registered and withdrew from the course.

Academic Misconduct Policy

All students are required to read, know, and comply with the Johns Hopkins University Krieger School of Arts and Sciences (KSAS) / Whiting School of Engineering (WSE) Procedures for Handling Allegations of Misconduct by Full-Time and Part-Time Graduate Students.

This policy prohibits academic misconduct, including but not limited to the following: cheating or facilitating cheating; plagiarism; reuse of assignments; unauthorized collaboration; alteration of graded assignments; and unfair competition. Course materials (old assignments, texts, or examinations, etc.) should not be shared unless authorized by the course instructor. Any questions related to this policy should be directed to EP’s academic integrity officer at ep-academic-integrity@jhu.edu.

Students with Disabilities - Accommodations and Accessibility

Johns Hopkins University values diversity and inclusion. We are committed to providing welcoming, equitable, and accessible educational experiences for all students. Students with disabilities (including those with psychological conditions, medical conditions and temporary disabilities) can request accommodations for this course by providing an Accommodation Letter issued by Student Disability Services (SDS). Please request accommodations for this course as early as possible to provide time for effective communication and arrangements.

For further information or to start the process of requesting accommodations, please contact Student Disability Services at Engineering for Professionals, ep-disability-svcs@jhu.edu.

Student Conduct Code

The fundamental purpose of the JHU regulation of student conduct is to promote and to protect the health, safety, welfare, property, and rights of all members of the University community as well as to promote the orderly operation of the University and to safeguard its property and facilities. As members of the University community, students accept certain responsibilities which support the educational mission and create an environment in which all students are afforded the same opportunity to succeed academically. 

For a full description of the code please visit the following website: https://studentaffairs.jhu.edu/policies-guidelines/student-code/

Classroom Climate

JHU is committed to creating a classroom environment that values the diversity of experiences and perspectives that all students bring. Everyone has the right to be treated with dignity and respect. Fostering an inclusive climate is important. Research and experience show that students who interact with peers who are different from themselves learn new things and experience tangible educational outcomes. At no time in this learning process should someone be singled out or treated unequally on the basis of any seen or unseen part of their identity. 
 
If you have concerns in this course about harassment, discrimination, or any unequal treatment, or if you seek accommodations or resources, please reach out to the course instructor directly. Reporting will never impact your course grade. You may also share concerns with your program chair, the Assistant Dean for Diversity and Inclusion, or the Office of Institutional Equity. In handling reports, people will protect your privacy as much as possible, but faculty and staff are required to officially report information for some cases (e.g. sexual harassment).

Course Auditing

When a student enrolls in an EP course with “audit” status, the student must reach an understanding with the instructor as to what is required to earn the “audit.” If the student does not meet those expectations, the instructor must notify the EP Registration Team [EP-Registration@exchange.johnshopkins.edu] in order for the student to be retroactively dropped or withdrawn from the course (depending on when the "audit" was requested and in accordance with EP registration deadlines). All lecture content will remain accessible to auditing students, but access to all other course material is left to the discretion of the instructor.