Math 221 Course Webpage
Linear Algebra and Applications
Fall 2024, Duke University
General information |
Course description |
Lecture and covid policies |
Assignments |
Homework and lecture schedule |
Grading |
Links |
Fine print
Goal: Students will become proficient in both using and
understanding the theory and algorithms of linear
algebra and will learn how to write rigorous
mathematical arguments.
Lectures: Tuesdays and Thursdays
11:45 – 13:00, Physics Building 259
Text: Linear Algebra: A Geometric Approach,
by Ted Shifrin and Malcolm Adams, second edition.
In past semesters, this textbook has been available for 3-hour
checkout at the Duke Libraries; search the Libraries'
Top Textbooks program.
Contact information for the Instructor
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
— this is the primary mode of contacting me!
Zoom: https://duke.zoom.us/my/ezra.miller
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/221/221.html
Canvas site:
available to registered students via Duke NetID
Office hours: Tuesday, 13:00 – 14:15
Thursday, 13:00 – 14:15
Help Room
Thursdays, 19:00 - 22:00.
See here
for more information
Emergency procedures
What to do in an emergency
Duke|ALERT
Downloadable classroom emergency checklist
Goal: Students will become proficient in both using and
understanding the theory and algorithms of linear
algebra, and they will learn how to write rigorous
mathematical arguments.
Course content: Chapters 1 – 7 of the course text, by
Shifrin & Adams
Most items constitute one lecture
each; some fill two lectures:
- vectors
- n-dimensional geometry
- Gaussian elimination
- linear systems
- matrix algebra
- linear transformations
- elementary matrices; transpose
- linear subspaces
- linear independence
- bases; dimension
- abstract vector spaces
- inner products
- projections; least squares
- orthonormal bases; Gram-Schmidt algorithm
- changes of basis
- abstract linear transformations
- determinants
- formulas for determinants
- eigenvalues and eigenvectors
- spectral theorem
- Jordan form
- systems of ordinary differential equations (ODE)
Prerequisite: Second-semester calculus (Math 122, 112L, or 122L)
- (This item is a hold-over from the pandemic. But the overarching
message of respect for health and safety is still valid.)
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, if required.
- 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!
-
Recordings of lectures from Fall 2020 are posted on the
Math Department video server. Each video has been edited to
delete identifying information of students whose faces (and,
alas, names) occasionally appeared onscreen; if you detect
identifying information that accidentally remains, please
inform the instructor.
- Students are expected to attend lecture live, in person, unless
circumstances require lecture to be delivered virtually, in
which case attendance online is expected. There is currently
no plan to enable hybrid lectures.
- 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
(e.g., illness, athletic participation, religious holidays)
is not penalized.
- Lectures might sometimes need to be held online or with a recording.
- There will usually be days of advance warning if this step
needs to be taken, although in case of instructor illness
advance warning might not be possible.
- Links for these online sessions will be available on the
course Canvas page.
- These links are not to be shared. Sharing can
result in class disruptions from unwanted third parties and
can constitute a breach of class participants' privacy.
- Your video should remain on by default during any
virtual synchronous 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.
- That said, there are valid reasons to turn your video off,
particularly if you have issues with your internet
connection, your working space, or other external factors.
Turn off your video if you must, but in general everyone
should do their best to participate visually.
- Weekly homework will be collected via Canvas
- There will be two in-class midterm exams:
- Thursday 10 October
- Thursday 21 November
- There will be a final exam:
- 9:00 – 12:00 Thursday 12 December
- Quizzes may be given if homework performance lags.
Policies regarding graded work
- Due dates for the homework assignments this semester are
listed in the table below.
- You should work on the homework for a section of the book
immediately after the class in which it is covered.
- Homework for the book sections covered in a week will
be due before midnight on Thursday the following week.
- Late homework will not be accepted.
- Submit your homework solutions using Canvas
"Assignments":
Math 221 Canvas –>
Assignments
- Missed quizzes or exams: in general there will be no make-ups,
but some accommodation may be possible in one of the following
four situations: personal
emergencies or
tragedies, an incapacitating
illness, a
religious holiday, or varsity
athletic participation. Please visit these web-pages now
to familiarize yourself with the procedures.
- Oral collaboration on homework is encouraged while you
discuss the search for solutions, but when it comes time to
write them down,
- The written work you submit 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, you must indicate—on the
homework page—who your collaborators were.
- Collaboration of any sort on quizzes, in-class exams, or the
final exam is not permitted: you must work completely
independently, without collaborating or consultating human,
electronic, or written sources (including but not limited to
books or notes) and without giving or receiving help from
others. Portable electronic devices that are visible or
audible during exams will be confiscated until the exam period
ends. 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 dismissal from the University.
Again, don't test me.
- 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. Every case
brought by the instructor has resulted in sanctions: I have
never lost a case. 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.
- Consulting generative AI is prohibited for all aspects
of this course, including homework, midterms, and (if
applicable) term projects. Instead of asking a computer, ask a
classmate or the instructor.
- Non-AI computation: You may use a non-AI computational aid,
such as a computer algebra system, for the homework but it is
recommended that you avoid it as much as possible, particularly
since electronic devices of all sorts are not allowed on
quizzes, in-class exams, and the final exam.
- All submitted work (including quizzes and exams, in addition to
homework) must be typeset or written neatly and legibly. On
exams, instead of erasing please use a single line crossout;
this can sometimes help your score, if what you crossed out
turns out to have been correct.
- If a question requires more than a single expression or
equation, then your response must be phrased in complete
English sentences. Writing mathematics requires as much
grammatical rigor as writing about literature or history.
Would you write a sentence without a subject or a period in a
Philosophy essay?
- The logic of a response to a homework or exam question must be
completely clear for full credit.
Math 721 vs. Math 221: graduate coursework
- If you registered for Math 721 instead of Math 221, then
in addition to all of the other work listed on this
syllabus, you must submit a term paper of roughly 8 - 10
pages. Parameters for this assignment must be discussed
with the instructor starting no later than Fall break.
- Math 721 has an altered grading scheme to take the
additional graduate coursework into account.
- Math 721 students are encouraged to contact the instructor as
early in the semester as possible to discuss the altered
grading scheme as well as the term project.
This schedule reflects the current plan, but it is subject to change.
The exam schedule, however, is fixed.
If a lecture or assignment hasn't been posted, and you think it should
have been, then please do email me. Sometimes I encounter problems
(such as, for example, the department's servers going down) while
posting assignments; other times, I might simply have forgotten to
copy the updated files into the appropriate directory, or to set the
permissions properly.
Read and study the text carefully before attempting the assignments.
Make sure you fully understand the given proofs and examples. Note
that there are examples in the text similar to most of the homework
problems. The material in gray shaded boxes consists of definitions;
learn them precisely. (Often when students say they do not know how
to do a problem it is because they don't know the definitions of the
terms in the problem.) The material in blue shaded boxes introduces
points of logic and techniques of proof that you will find helpful in
writing your arguments. If you have trouble understanding something
in the text after working on it for a while, then see me in office
hours or e-mail me.
You can download
all lectures in one PDF file.
Alternatively, for notes on an individual lecture, click on the date
in the table.
You can view
all of the lecture videos on one webpage.
Alternatively, for an individual lecture video, click on the lecture #
in the table.
(Note: there are no videos for Lectures 25 and 26, since the pandemic
semester was one week shorter than usual.)
#/Video |
Date |
Sections |
Topic |
Assignment due Thursday unless otherwise noted |
1. |
Tue 27 Aug |
1.1 – 1.2 |
vectors |
|
2. |
Thu 29 Aug |
1.2 – 1.3 |
n-dimensional geometry |
|
3. |
Tue 3 Sep |
1.4 |
matrix multiplication |
|
4. |
Thu 5 Sep |
1.4 |
Gaussian elimination |
1.1: 6(a,c,g), 7, 8, 9, 21, 22, 23, 25, 29
1.2: 1(b,d,g), 2(b,d,g), 4, 9, 11, 13 (no geometric
interpretation necessary), 16, 18 |
5. |
Tue 10 Sep |
1.5 |
linear systems |
|
6. |
Thu 12 Sep |
1.5, 1.6.1 |
solving linear systems |
1.3: 1(a,c,f), 3(a,d,e), 5, 8, 10, 12
1.4: 1, 3(a–f), 4(d,f), 10, 11, 12, 13, 15 |
7. |
Tue 17 Sep |
2.2 – 2.3 |
linear transformations |
|
8. |
Thu 19 Sep |
2.1 – 2.2 |
matrix algebra |
1.5: 1, 2(a,b), 3(a,c), 4a, 6, 10, 12, 13, 14
1.6: 5, 7, 9, 11 |
9. |
Tue 24 Sep |
2.4 – 2.5 |
elementary matrices
transpose |
|
10. |
Thu 26 Sep |
3.1 – 3.2 |
linear subspaces |
2.1: 1(a,c,f), 2, 5, 6, 7, 8, 12(a,b,d), 14
2.2: 5, 7, 8
2.3: 1(b,d,f), 2(a,c,d), 4, 8, 11, 13(a,b,c), 16 |
11. |
Tue 1 Oct |
3.2 |
linear subspaces |
|
12. |
Thu 3 Oct |
3.3 |
linear independence |
2.4: 7, 12
2.5: 1(a,f,j), 4, 8, 9, 12, 15, 19(a,b), 22, 23
3.1: 1, 2(a,c,d), 6, 9(b,c), 10, 12, 13, 14 |
13. |
Tue 8 Oct |
3.3 – 3.4 |
bases; dimension |
|
|
Thu 10 Oct |
|
FIRST MIDTERM EXAM |
Due Wed: 3.2: 1, 2(a,b), 10, 11
3.3: 1, 2, 8, 10, 11, 14, 15, 19, 21, 22 |
|
Tue 15 Oct |
|
no class: Fall break |
|
14. |
Thu 17 Oct |
3.4 |
bases; dimension |
|
15. |
Tue 22 Oct |
3.6 |
abstract vector spaces |
|
16. |
Thu 24 Oct |
3.6 – 4.1 |
inner products; projections |
3.3: 5(a,b)
3.4: 3(a,b,d), 4, 8, 17, 20, 24 |
17. |
Tue 29 Oct |
4.1 – 4.2 |
least squares; orthonormal bases; Gram-Schmidt |
|
18. |
Thu 31 Oct |
4.4 |
abstract linear transformations |
3.6: 1, 2(a,c,d), 3(a,c,f), 4, 6(a,b), 9, 13, 14(b,c), 15(a,b) |
19. |
Tue 5 Nov |
4.3 |
change of basis |
|
20. |
Thu 7 Nov |
4.3 – 4.4 |
review of change of basis |
4.1: 1(a,b), 3, 6, 7, 9, 11, 13, 15
4.2: 2(b,c), 3, 6, 7(a,b), 8a, 9a, 11, 12(a,b) |
21. |
Tue 12 Nov |
5.1 |
determinants |
|
22. |
Thu 14 Nov |
5.2 |
formulas for determinants |
4.4: 2, 5, 8, 14, 16, 22(a,b),
additional problems 1–9
4.3: 3, 9, 12, 19,
additional problems 1–3 |
23. |
Tue 19 Nov |
6.1 |
eigenvalues and eigenvectors |
|
|
Thu 21 Nov |
|
SECOND MIDTERM EXAM |
Due Wed: 5.1: 1(a,b,c), 2, 3, 4, 7, 9(a), 10, 11
5.2: 1a, 3, 4, 5(a,c,f), 7, 8, 10 |
24. |
Tue 26 Nov |
6.2 |
diagonalizability
spectral theorem |
|
|
Thu 28 Nov |
|
no class: Thanksgiving |
Due Wed: 4.3: 18, 20, 21
6.1: 1 (do as many as you can stand!), 2, 3, 4, 6, 10, 12, 14 |
25. |
Tue 3 Dec |
7.1; 6.4 |
Jordan form |
|
26. |
Thu 5 Dec |
7.3 |
matrix exponentials; systems of ODE |
6.2: 1 (do as many as you can stand!), 3, 4, 6, 11, 16(a-c)
6.4: 1, 2, 3, 4, 5, 8, 10, 11, 13 |
|
Thu 12 Dec |
|
FINAL EXAM, 9:00 – 12:00 |
*optional* 7.1: 4, 6, 7, 8, 14, 16
(but you will be
*optional* 7.3: 1, 4, 5, 8, 9, 10, 13, 14 tested on
this material)
|