Math 544H, Spring 2025
Linear Algebra, Honors -- Math 544H
Frank Thorne - Spring 2025
University of South Carolina
Instructor: Frank Thorne, LeConte 447, thorne [at] math [dot] sc [dot] edu
Office Hours: Tuesdays 9:30-11:00 and Wednesdays 3:30-5:00.
Course objectives/learning outcomes:
Successful students will:
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Understand the concept of a real vector space from an algebraic and a geometric point of view. The student will master related concepts such as bases and linear independence, norms, inner products, and so forth.
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Understand the most important maps between vector spaces -- the
linear transformations. The successful student will be able to describe
their structure and describe it in terms of ranks, nullspaces, etc.
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Understand how to manipulate matrices and explain the correspondence between
matrices and linear transformations.
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Understand what eigenvalues are, why they are interesting, and why one would
want to compute them.
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See applications of all of the above to engineering and physical and social sciences. To give just one example --- at the heart of Google search is an eigenvalue. (The student
will not master this over the course of only one semester! If you give it thirty years, maybe. But people are developing new applications for linear algebra all the time.)
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Develop their ability to understand and explain rigorous mathematics and prove simple statements. This is a 500-level math class, and as such proofs will be required on homeworks and exams. If you have limited background in proofs, then this course will help bolster it.
Course non-objectives/learning non-outcomes:
Linear algebra is too big of a subject to really do justice to in one semester. Therefore we will mostly omit some beautiful topics.
- The course will probably say little about the very interesting and important questions
when doing linear algebra in the "real world" -- which involves imprecise measurements and computational precision issues. This beautiful topic will be emphasized in other courses.
- The course will develop only a minimal amount of formalism -- so the vector space axioms, fields other than R, infinite-dimensional vector spaces, inner products other than the "obvious" one coming from geometry, etc. will not be talked about much. This material would be more appropriate in a more advanced course.
Please see me at the end of the term, or pursue a relevant term project, if you would be interested in learning more about these aspects of the subject.
Warning. You should expect 5-8 hours of homework a week in this class, which is more than most other instructors assign;
in my experience there is no other way to learn the material. Your consistent effort will certainly lead to improved understanding,
and it will almost certainly lead to you earning high grades.
Text : Jim Hefferon,
Linear Algebra (fourth edition).
The book is available by clicking the link above, or directly here.
It is completely free. You can also purchase a cheap printed copy if you like.
Thanks to the author for making this high quality material available for free! If you enjoy his books,
please consider contacting them directly and saying so.
Lectures : 9:40-10:30, MWF, LC 348.
Exam schedule : The exams will be take home and pledged, of two hours duration.
You will have at least a 72 hour window to complete them, at a time of your choosing.
Dates will
be announced at least a week in advance.
Homework : Warning. I assign a lot of homework.
The homework is intended to take 5-8 hours a week. That is a lot. Please
count on making a consistent effort to do well in this class! Starting the night before is a bad idea.
If homework takes you more than 10 hours on any given week, then that is more than I intended;
please let me know.
There will also be at least one bonus problem on each homework.
Please note. You will be graded both on correctness and on quality of exposition. Indeed, a major
focus of Math 544 is the ability to communicate mathematical ideas clearly.
The standard is that someone who doesn't know the answer should be able to easily follow your work.
In particular, please write in complete English sentences and draw clear diagrams where appropriate.
Any work that is confusing, ambiguous, or poorly explained will not receive full credit.
Final project :
Instead of a final exam, this course will ask you to do a final project on some aspect of linear
algebra. The choice of topic is up to you; possible topics include: theoretical aspects of linear algebra;
computational aspects of linear algebra; connections to other areas of math; applications to statistics,
computer science, engineering, and the natural or social sciences.
The project will consist of an initial proposal, a term paper, and a 20-minute presentation during the final exam period.
Group work is encouraged. More details will be announced.
Grading scale: You are guaranteed at least the following grades: A = 88+, B+ = 83+, B = 75, C+ = 70+, C = 60+, D = 50+.
| Grade component |
% of grade |
| Two midterm exams |
20% x 2 |
| Final project: |
20% |
Homework: |
40% |
Contacting me :
Please contact me if you have any questions about the course, about my expectations, about my lectures,
about the homeworks, about the reading, or about anything else. The syllabus is demanding and it is my
job to help you succeed.
Policies :
Late homework may be accepted, especially if arranged in advance, but no guarantees are made, and repeated late homeworks will be accepted only in case of severe extenuating circumstances.
Academic honesty is expected of all students.
Attendance is encouraged, and you are responsible for material covered in class,
but I will not be taking roll or enforcing any penalty for absences.
If you have any disabilities that require accommodation, please register with
the Student Disability Resource Center.
It is my goal to create a welcoming classroom environment, free of racism, sexism, homo- or transphobia, discrimination, bullying, insults, or harassment. Please bring any concerns to me; major or repeated violations will be reported to the Office of Student Conduct.