Math 221 Course Webpage
Linear Algebra and Applications
Fall 2021, 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
12:00 – 13:15, Physics Building 119 (Section 02)
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
Office hours: Monday, 16:15 – 17:15
Thursday, 13:15 – 14:30
Help Room
Mondays and Thursdays, 19:00 - 22:00 and other times.
See here
for more information
Emergency procedures
What to do in an emergency
Duke|ALERT
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)
- 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!
- Students are expected to attend lecture in
person.
- Attendance in lecture is a baseline for class participation
grades, which can enhance or detract from homework grades,
although earning real credit along these lines requires
contribution to in-class discourse.
- Occasional absence for expected sorts of circumstances are
of course not penalized.
- Students registered for in-person lectures can
watch a recorded lecture online
if they are quarantined or otherwise incapacitated.
Policies regarding covid
- Students who test positive for covid are required to contact
the Duke Student Health
COVID Hotline at (919) 684-1258. They will advise you on
quarantining and start the contact tracing process. A positive
test is a medical diagnosis; as such it is not required that
students directly inform faculty or fellow students directly.
If a determination is made that faculty or students have been
exposed, then that will be borne out by contact tracing, which
will result in the exposed individuals being informed.
- If you have to quarantine then you should notify your
professors using the short-term illness notification form
(STINF). You are free to divulge your covid testing status if
you wish, but informing your professor that you aren't coming
to class because of a positive covid test is not sufficient;
you must submit a short-term illness notification form.
- Students who have been exposed to someone affiliated with Duke
(e.g., a student or faculty member) should receive
communication from Student Health as part of contact tracing.
If you were informed about exposure to a Duke student by the
student instead of by a contact-tracing communication, then
PLEASE contact the student who tested positive and insist that
they contact the Duke
Student Health COVID Hotline at (919) 684-1258 as required.
- You can reach Student
Health at (919) 681-9355. If you have been exposed, then they
will give you more information when you call, and you will be
required to call the Duke Student Health COVID Hotline at (919)
684-1258.
- General info about students testing positive:
- All students who test positive for COVID-19 are required to
isolate for 10 days as recommended by CDC and Duke medical
advisors.
- Students who reside on campus in Duke housing will be moved
to an isolation facility on campus.
- Students who reside off campus must isolate in their own
residence.
- Students who are identified by contact tracing will be
subject to more frequent testing for the period after their
exposure but are no longer required to quarantine as
recommended by CDC.
- All students who are isolated due to a positive COVID-19
test are monitored by Duke Student Health.
- Weekly homework will be collected and graded via Gradescope
- There will be two in-class midterm exams:
- Thursday 30 September
- Thursday 18 November
- There will be a final exam:
- 14:00 – 17:00 Monday 13 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 at
12:00, the start of the following Tuesday's class.
- Late homework will not be accepted.
- 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.
- 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, 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 books or notes and without giving or
receiving help from others.
- Students are expected to adhere to the
Duke Community Standard. Students affirm
their commitment to uphold the values of the Duke University
community:
- 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.
You must reaffirm your committment to these standards by your
signature on all work.
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. 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. Most
infractions result in the student receiving a score of zero for
that assignment, and the instructor reserves the right to
further reduce the final grade for the course by one or more
letter grades—possibly to a failing grade—at the
discretion of the instructor.
- Computer policy: You may use a computational aid for the
homework but I recommend avoiding it as much as possible,
particularly since electronic devices of all sorts are not
allowed on quizzes, in-class exams, and the final exam.
Portable electronic devices that are visible or audible during
exams will be confiscated until the exam period ends.
- 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 sentences.
- The logic of a proof must be completely clear for full credit.
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.)
# |
Date |
Sections |
Topic |
Assignment due |
1. |
Tue 24 Aug |
1.1 – 1.2 |
vectors |
|
2. |
Thu 26 Aug |
1.2 – 1.3 |
n-dimensional geometry |
|
3. |
Tue 31 Aug |
1.4 |
matrix multiplication |
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 |
4. |
Thu 2 Sep |
1.4 |
Gaussian elimination |
|
5. |
Tue 7 Sep |
1.5 |
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 |
6. |
Thu 9 Sep |
1.5, 1.6.1 |
solving linear systems |
|
7. |
Tue 14 Sep |
2.2 – 2.3 |
linear transformations |
1.5: 1, 2(a,b), 3(a,c), 4a, 6, 10, 12, 13, 14
1.6: 5, 7, 9, 11 |
8. |
Thu 16 Sep |
2.1 – 2.2 |
matrix algebra |
|
9. |
Tue 21 Sep |
2.4 – 2.5 |
elementary matrices
transpose |
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 |
10. |
Thu 23 Sep |
3.1 – 3.2 |
linear subspaces |
|
11. |
Tue 28 Sep |
3.2 |
linear subspaces |
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 |
|
Thu 30 Sep |
|
FIRST MIDTERM EXAM |
|
|
Tue 5 Oct |
|
no class: Fall break |
|
12. |
Thu 7 Oct |
3.3 |
linear independence |
|
13. |
Tue 12 Oct |
3.3 – 3.4 |
bases; dimension |
3.2: 1, 2(a,b), 10, 11
3.3 1, 2, 8, 10, 11, 14, 15, 19, 21, 22 |
14. |
Thu 14 Oct |
3.4 |
bases; dimension |
|
15. |
Tue 19 Oct |
3.6 |
abstract vector spaces |
3.3: 5(a,b)
3.4: 3(a,b,d), 4, 8, 17, 20, 24 |
16. |
Thu 21 Oct |
3.6 – 4.1 |
inner products; projections |
|
17. |
Tue 26 Oct |
4.1 – 4.2 |
least squares; orthonormal bases; Gram-Schmidt |
3.6: 1, 2(a,c,d), 3(a,c,f), 4, 6(a,b), 9, 13, 14(b,c), 15(a,b) |
18. |
Thu 28 Oct |
4.4 |
abstract linear transformations |
|
19. |
Tue 2 Nov |
4.3 |
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) |
20. |
Thu 4 Nov |
4.3 – 4.4 |
review of change of basis |
|
21. |
Tue 9 Nov |
5.1 |
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 |
22. |
Thu 11 Nov |
5.2 |
formulas for determinants |
|
23. |
Tue 16 Nov |
6.1 |
eigenvalues and eigenvectors |
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 |
|
Thu 18 Nov |
|
SECOND MIDTERM EXAM |
|
24. |
Tue 23 Nov |
6.2 |
diagonalizability
spectral theorem |
4.3: 18, 20, 21
6.1: 1 (do as many as you can stand!), 2, 3, 4, 6, 10, 12, 14 |
|
Thu 25 Nov |
|
no class: Thanksgiving |
|
25. |
Tue 30 Nov |
7.1; 6.4 |
Jordan form |
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 |
26. |
Thu 2 Dec |
7.3 |
matrix exponentials; systems of ODE |
|
|
Mon 13 Dec |
|
FINAL EXAM, 14:00 – 17:00 |
*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)
|