General information | Course description | Lecture notes | Lecture and video policies | Assignments | Homework schedule | Grading | Links | Fine print

**Lectures:** Tuesday and Thursday, 12:00 – 13:15, Physics Building 154

**Textbooks:**

*Linear Algebra and its Applications*, by Peter D. Lax, second edition (book; available electronically)- Kristopher Tapp:
*Matrix Groups for Undergraduates*(book; available electronically) - G. W. Stewart and Ji-guang Sun:
*Matrix Perturbation Theory*(book; not electronic)

**Online resources:**

- Climenhaga: Lecture notes for Advanced Linear Algebra
- Hefferon:
*Linear Algebra* - Treil: Linear Algebra Done Wrong
- Cherney, Denton, and Waldron: Linear Algebra
- Cornell: Honors Linear Algebra
- Gustafson: Advanced Topics in Linear Algebra (not a full text, but a few pages of notes)

**Name:** Professor Ezra Miller
**Address:** Mathematics Department,
Duke University, Box 90320,
Durham, NC 27708-0320
**Office:** Physics 209
**Phone:** (919) 660-2846
**Email:** *ezramath.duke.edu*
**Webpage:**
*https://math.duke.edu/people/ezra-miller*
**Course webpage:** you're already
looking at it... but it's
*https://services.math.duke.edu/~ezra/403/403.html*
**Sakai site:** available to registered students via Duke NetID
**Zoom links:** available to registered students via Sakai
**Office hours:** Monday, 16:15 – 17:15 on Zoom

Thursday, 13:15 – 14:30 on Zoom from
Physics 209 or outside

- abstraction
- approximation
- variation
- positivity
- convexity
- application

Students will continue to develop their skills in mathematical exposition, both written and oral, including proofs.

**Course content:**

- basic algebra: arbitrary fields, quotients, exact sequences, duality
- normed linear spaces
- convexity
- grassmannians and flags
- Hermitian, positive (semi-)definite, and normal matrices
- principal component analysis (PCA) and singular value decomposition (SVD)
- classical matrix groups
- perturbation of eigenvalues
- Perron-Frobenius theory: matrices with positive entries
- tensors and exterior algebra

- Lecture 1: basic algebra of groups, arbitrary fields, isomorphisms
- Lecture 2: quotients, exact sequences, homology, Euler characteristic
- Lecture 3: duality, complex numbers, Hermitian inner products
- Lecture 4: Jordan form, Fundamental Thm of Algebra, direct sum, minimal polynomial, Cayley-Hamilton Thm
- Lecture 5: Banach spaces: norms, topology, equivalence
- Lecture 6: convex set, polytope, interior, boundary, separating hyperplane
- Lecture 7: supporting hyperplane, extreme point, Krein-Milman Thm
- Lecture 8: grassmannians as quotients and as manifolds
- Lecture 9: manifolds (recap), flag varieties
- Lecture 10: isometries of spheres and real inner product spaces
- Lecture 11: unitary matrices, upper-triangular representations, spectral theorem, normal operators
- Lecture 12: positive (semi)definite operators, singular values, polar and Schmidt decompositions, singular value decomposition (SVD)
- Lecture 13: ellipsoids, operator and Frobenius norms, principal component analysis (PCA)
- Lecture 14:
Lie algebra as tangent space, computations for GL
_{n}, SL_{n}, O_{n} - Lecture 15: matrix exponentials, Lie groups as manifolds
- Lecture 16: matrix norms, consistency, general perturbation theorems
- Lecture 17: Bauer-Fike theorem, Gerschgorin theory
- Lecture 18: entrywise positive matrices, dominant eigenvalue, Perron's theorem
- Lecture 19: directed graphs, stochastic matrices, Markov chains, Frobenius theorem
- Lecture 20: multilinear algebra, tensor product, universal property, category
- Lecture 21: alternating operator, exterior power, volume, determinant, functor
- Lecture 22: cross product, Laplace expansion, minor, r-dimensional volume, Plücker coordinate

- Everyone attending class in person, including students and the
lecturer, must comply with the Duke Compact
regarding health and safety of yourself and others.
- Keep your mask on at all times.
- Keep your distance from others: six feet. Even when you are entering or exiting the classroom; don't gather by the door.
- Failure to comply will result in immediate dismissal from the classroom. In particular,
- kindly remind the lecturer if they are not in compliance, and dismiss them if they refuse to comply!

- Lectures will be live-streamed via Zoom. Links for these Zoom
sessions are on the course Sakai page. These
**Zoom links are not to be shared**. - Students are expected to attend lecture live, either in person
or online as corresponds with their officially registered
section.
- Class participation can enhance or detract from homework grades. Attendance in lecture is a baseline for participation, which more substantially requires contribution to in-class discourse.
- Occasional absence for expected sorts of circumstances are not penalized.
- Students registered for in-person lectures can attend live online if they are quarantined or otherwise incapacitated. However, in-person attendance is a privilege with limited availability; regular failure to attend physically in the classroom is grounds for losing your seat to another student.

- For online students, your video should remain on by default
during lecture.
- The classroom experience is a communal one. It is dependent on multi-way communication. You should be able to see the lecturer and your classmates, and they should be able to see you.
- There are reasons to turn your video off, particularly if you have issues with your internet connection, your working space, or other external factors. But in general everyone should do their best to participate visually.

- Lectures will be recorded and made available via the class
Sakai website.
- These recordings are required for rare circumstances in which students must temporarily participate in lecture asynchronously.
- Posting will ordinarily occur within 24 hours. It takes Zoom time to process video, it takes time to upload large files, and the instructor has daily responsibilities that might prevent immediate uploading.

- Students are not permitted to share any video of anybody else from any portion of the class, including the discussion sections, with parties outside of the class.
- In addition to potential accessibility issues in a typical academic year, remote learning may present additional challenges. You may be experiencing unreliable wi-fi, lack of access to quiet study spaces, varied time zones, or additional responsibilities while studying at home. If you are experiencing these or other difficulties, please contact the instructor to discuss possible accommodations.

**Reading assignments**are included at the top of each homework and midterm.**Due dates**for the five homework assignments, two midterms, and final projects this semester are listed in the table below. The final project also includes an oral presentation near the end of the semester, during a class period to be determined.- All homework and midterm assignments will be take-home.
**Each is due at the time of day specified on the assignment**. **All solutions you turn in, including midterms, homework, and term projects, must be typewritten using the provided LaTeX template.**Communicating your ideas is an integral part of mathematics. In addition to the usual PDF files, LaTeX source files for each of the homework assignments as well as each of the midterms will be provided. You are required to use these as LaTeX templates for your solutions, by filling in your responses in those files. I am happy to answer any questions you might have about LaTeX, though you should consider asking your classmates first. The reason why**it is crucial for you to use the provided LaTeX templates is because they contain margins settings and commands required for grading**.**Submit your homework and midterm solutions using the "Drop Box" feature on Sakai**. This feature is new to the instructor, so we'll see how it goes. Submit at least your .tex file (the grader or I may comment on your TeX usage); if you include your .pdf file as well then it can serve as verification that your system produces the same output as ours do.**The final project consists of an oral presentation and a written paper**of 10 – 15 pages in length. The length of the oral presentation will depend on how many students are registered once the semester is fully under way.**The final projects are to be completed in pairs**. If you are unable to find a partner, then the instructor will assign you one.

Assignment | Due Date | Problems |
---|---|---|

Homework #1 |
Tue. 9 February | in PDF or LaTeX |

Homework #2 |
Tue. 23 February | in PDF or LaTeX |

Midterm 1 | Tue. 2 March | in PDF or LaTeX |

Homework #3 |
Tue. 16 March | |

Homework #4 |
Tue. 30 March | |

Homework #5 |
Tue. 13 April | |

Midterm 2 | Tue. 20 April | |

Final project | Thu. 29 April, 9am–noon |

- Late homework will not be accepted.
- The logic of a proof must be completely clear for full credit.
**You must cite sources in your solutions**. If you rely on so-and-so's theorem, then you must say where you found it. Be specific: "the dual rank theorem" is not precise; in contrast, "[Climenhaga, Theorem 5.10]" is. Theorems are often known by many names, so anyone grading your work could fail to recognize any given theorem by a name you might attach.**Students are expected to adhere to the Duke Community Standard**. Students affirm their commitment to uphold the values of the Duke University community by signing a pledge that states:- I will not lie, cheat, or steal in my academic endeavors;
- I will conduct myself honorably in all my endeavors;
- I will act if the Standard is compromised.

**The instructor has a long record of detecting and convicting violations**, resulting in a wide range of sanctions, from grade changes to removal from the University. Don't test me.**Collaboration on homework is encouraged**while you discuss the search for solutions, but when it comes time to write them down, the work you turn in must be yours alone: you are not allowed to consult anyone else's written solution, and you are not allowed to share your written solutions. (It is very easy to tell when solutions have been copied or written together.) If you collaborate while searching for solutions,**you must indicate—on the homework page—who your collaborators were**. Failure to identify your collaborators is a breach of the Duke Community Standard.- In contrast,
**no collaboration or consultation of human or electronic sources—except for the "Textbooks" and specific "Online resources" listed on this syllabus—is allowed for either of the two exams**. You must work completely independently, without giving or receiving help from others. It bears repeating that the instructor has a long record of detecting and convicting violations of the Duke Community Standard, resulting in a wide range of sanctions, from grade changes to removal from the University. Again, don't test me. - If a student is responsible for academic dishonesty on a graded item in this course, then the student will have an opportunity to admit the infraction and, if approved by the Office of Student Conduct, resolve it directly through a faculty-student resolution agreement; the terms of that agreement would then dictate the grading response to the assignment at issue. If the student is found responsible through the Office of Student Conduct and the infraction is not resolved by a faculty-student resolution agreement, then the sanctions can be severe and, depending on the nature of the infraction, might be out of the hands of the instructor, being instead determined by the Office of Student Conduct or another University body.

- 40% Homework and participation
- 15% Midterm #1
- 15% Midterm #2
- 30% Term project

- list of all courses given in the Math Department in Spring 2021.
- final exam schedule for all Duke courses.
- Official University calendar for academic year 2020–2021.
- Academic Resource Center (ARC) for Learning Consultations, Peer Tutoring, Learning Communities, ADHD/LD Coaching, Outreach Workshops, GRE/MCAT Prep, Study Connect, and more.
- Fundamental University policies concerning academic dishonesty and related matters.
- What to do in an emergency
- Duke|ALERT
- Duke Community Standard
- Duke Compact

Many thanks are due to Jeremy Martin and Vic Reiner, who provided templates for this webpage many years ago.

The views and opinions expressed in this page are strictly those of the page author. The contents of this page have not been reviewed or approved by Duke University.

*ezramath.duke.edu*

*Sat Feb 27 15:33:46 EST 2021*