Math 221 Course Webpage
Linear Algebra and Applications
Fall 2020, Duke University
General information |
Course description |
Lecture and video policies |
Homework schedule |
Goal: Students will become proficient in both using and
understanding the theory and algorithms of linear
algebra and will learn how to write rigorous
Lectures: Tuesdays and Thursdays
13:45 – 15:00, LSRC B101 "Love Auditorium" (Section 01)
13:45 – 15:00, online (Section 02)
Discussion sections (mandatory): depends on your selection at registration
Mondays 12:00 – 12:50 (Section 01D), Prof. Hain
Mondays 13:45 – 14:35 (Section 02D), Prof. Hain
Mondays 15:30 – 16:20 (Section 03D), Prof. Hain
Days and times TBD (Section 04D), Prof. Myer
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
— this is the primary mode of contacting me!
Course webpage: you're already
looking at it... but it's
Tuesdays and Thursdays, 15:00 – 16:30, by Zoom
Mondays and Thursdays, 19:00 - 22:00 and other times.
for more information, including Zoom links.
What to do in an emergency
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
Course content: Chapters 1 – 7 of the course text, by
Shifrin & Adams
Most items constitute one lecture
each; some fill two lectures:
Prerequisite: Second-semester calculus (Math 122, 112L, or 122L)
- 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
- formulas for determinants
- eigenvalues and eigenvectors
- spectral theorem
- Jordan form
- systems of ordinary differential equations (ODE)
- 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
- 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 and discussion sections will be live-streamed via Zoom.
- Students are expected to attend lecture live, either in person
or online as corresponds with their officially registered
- 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 attend live
online if they are quarantined or otherwise incapacitated.
- Your video should remain on by default during discussion
section and (if you are online) during lecture.
- The classroom experience, be it discussion or lecture, 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.
- During exams, your video is required to be on unless you have a
specific justification for turning it off. For reasons of
privacy you won't be asked to detail the justification,
but without video, alternate arrangements may need to be made
- Lectures will be recorded and uploaded to the class Sakai
- These recordings are required for rare circumstances in
which students must temporarily participate in lecture
- Posting will ordinarily occur within 24 hours. It takes
Zoom time to process video, it takes time to upload large
files, and the instructors have daily responsibilities that
might prevent them from uploading immediately.
- Nobody is permitted to share any video of anybody else from any
portion of the class, including the discussion sections, with
parties outside of the class.
Policies regarding graded work
- Weekly homework will be collected and graded via Gradescope
- There will be two in-class midterm exams:
- Tuesday 29 September
- Tuesday 10 November
- There will be a final exam:
- 14:00 – 17:00 Sunday 22 November
- Quizzes may be given if homework performance lags.
- 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
17:00, two hours after 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
tragedies, an incapacitating
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. You must reaffirm your
committment to these standards on all work.
- 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 student will receive 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
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.
||Tue 18 Aug
||1.1 – 1.2
||Thu 20 Aug
||1.2 – 1.3
||Tue 25 Aug
||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
||Thu 27 Aug
||Tue 1 Sep
||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
||Thu 3 Sep
||solving linear systems
||Tue 8 Sep
||2.2 – 2.3
||1.5: 1, 2(a, b), 3(a, c), 4a, 6, 10, 12, 13, 14
1.6: 5, 7, 9, 11
||Thu 10 Sep
||2.1 – 2.2
||Tue 15 Sep
||2.4 – 2.5
|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
||Thu 17 Sep
||3.1 – 3.2
||Tue 22 Sep
||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 24 Sep
||Tue 29 Sep
||FIRST MIDTERM EXAM
||3.2: 1, 2(a,b), 10, 11
||Thu 1 Oct
||3.3 – 3.4
||Tue 6 Oct
||3.3 1, 2, 5(a,b), 8, 10, 11, 14, 15, 19, 21, 22
||Thu 8 Oct
||abstract vector spaces
||Tue 13 Oct
||3.6 – 4.1
||inner products; projections
||3.4: 3(a, b, d), 4, 8, 17, 20, 24
3.6: 1, 2(a, c, d), 3(a, c, f), 4, 6(a, b), 9, 13, 14(b, c), 15(a, b)
||Thu 15 Oct
||4.1 – 4.2
||least squares; orthonormal bases; Gram-Schmidt
||Tue 20 Oct
||abstract linear transformations
||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)
||Thu 22 Oct
||change of basis
||Tue 27 Oct
||more change of basis
||4.4: 2, 5, 8, 11, 13, 14,
16, 22(a,b), additional problems 1–9
||Thu 29 Oct
||Tue 3 Nov
||formulas for determinants
||4.3: 3, 7, 9, 12, 18, 19, 20, 21, additional problems 1–3
5.1: 1(a, b, c), 2, 3, 4, 7, 9(a), 10, 11
||Thu 5 Nov
||eigenvalues and eigenvectors
||Tue 10 Nov
||SECOND MIDTERM EXAM
||5.2: 1a, 3, 4, 5(a,c,f), 7, 8, 10
6.1: 1 (do as many as you can stand!), 2, 3, 4, 6, 10, 12, 14
||Thu 12 Nov
||diagonalizability; spectral theorem
||Mon 16 Nov
||(perhaps: Jordan form; ODE systems)
||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
||Sun 22 Nov
||FINAL EXAM, 14:00 – 17:00
||*optional* 7.1: 4, 6, 7, 8, 14, 16
*optional* 7.3: 1, 4, 5, 8, 9, 10, 13, 14