625.609.81 - Matrix Theory

Expanded Course Description

In this course, topics include the methods of solving linear equations, Gaussian elimination, triangular factors and row exchanges, 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. The course also covers applications to statistics (least squares fitting to linear models, covariance matrices) and to vector calculus (gradient operations and Jacobian and Hessian matrices). MATLAB software (or equivalent) will be used in some class exercises.

Course Structure

The course materials are divided into 12 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 discussions 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
• 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)

• Solve systems of linear equations using Gaussian Elimination
• Compute and differentiate between the fundamental matrix decompositions (LU, QR, eigenvalue decomposition, and SVD)
• Construct proofs of simple theorems
• Apply sophisticated theorems to solve problems in matrix theory
• Calculate the coordinate representation of elements of an abstract vector space in various bases, and differentiate between the abstract vectors and their coordinate representations.
• Calculate the condition number of a matrix and show the impact a large condition number can have on the solutions to a problem.
• Explain the difficulties that may arise due to the numerical limitations of solving problems on a computer
• Solve matrix theory problems using a computer algebra system (such as MATLAB, Octave, Mathematica, Maxima, or others)
• Discuss the uses of matrix theory in a specific application.

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.

Other Materials & Online Resources

Noble, B., & Daniel, J. W. (1998). Applied Linear Algebra, third edition. Upper Saddle River, NJ: Prentice Hall, Inc.
ISBN-10: 0130412600
Note: This book is out of print. Any required reading from this book will be provided on Canvas via EReserves.

Nicholson, Keith (2019) Linear Algebra with Applications (Version 2019 - Revision A) Lyryx Open Texts.
Note: A digital copy of this book is available for free through https://lyryx.com/linear-algebra-applications/
.

Student Coursework Requirements

It is expected that each module will take approximately 10–15 hours per week to complete. Here is an approximate breakdown: 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 (approximately2-4 hours per week), group activities (approximately 1 hour per week), and problem set assignments (approximately 5–6 hours per week).

This course will consist of 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 the course. 

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 grade assignments within one week after assignment due dates. Grades for assignments can be viewed on Canvas.

Unless you have a previously arranged extension or there are exceptional extenuating circumstances, late submissions are generally not accepted without an extension. Students receive two 24-hour extensions by default. Please email me if you need additional extensions. 

Group Discussions (15% of Final Grade Calculation)
Group discussions 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 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

1. Compliment- Praise a specific aspect of the post. For example: "I like that your solution..."
2. Connect - Build connections to other relevant course materials. For example: "I had a similar observation/fi nding that..." or "This seems to relate to X from the lecture/reading/Module Y..."
3. 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..."
4. 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..."

Group activity posts are graded primarily based on effort and completeness (more so 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 use of the 3C+Q method, engagement with peers, and furthering the discussion. You may receive partial credit on late group activity posts. 

Grades and feedback for Group Activities will be through Canvas. For information on how to view feedback on group activities, 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 Papers (15% of Final Grade Calculation)
There are two applications papers: the first taking place in Modules 4-5 and the second taking place in Modules 10-11. Each of these requires students to pick a specific application of matrix theory that interests them, research it and compose a 2-3 page summary.

Exams (40% of Final Grade Calculation; 20% from the Midterm Exam, 20% from the Final Exam)
There are two exams: a Midterm in Module 6 and a cumulative Final in Module 12. 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 Canvas. Grades for exams can be viewed on Canvas.

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 submitted.

Effectively communicating mathematics through proof-writing is a significant focus of this course. In addition to logicaland 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.

EP uses a +/- grading system (see “Grading System”, Graduate Programs catalog, p. 10).

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

Course Evaluation

Assignments are due according to the dates posted in your Canvas course site. You may check these due dates in the Course Calendar or the Assignments in the corresponding modules. I will post grades one week after assignment due dates.

Writing clear, cogent proofs is an essential aspect of this course. As such, I will be much more particular about your writing style than in most mathematics courses. Egregious violations of the rules of the English language will detract from your score, as will logical, mathematical and notational inaccuracies.

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. These criteria apply to both undergraduates and graduate students taking the course.

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.