Spring 2019

Harvard Extension School

Mathematics E-21b - Spring 2019

Linear Algebra

Robert Winters

Course Assistants:
Jeremy Marcq
Renée Chipman

Handy Links:

Calendar of topics and
homework assignments

Linear Algebra Toolkit

Math E-21b syllabus

requires username/password

Canvas Site

Recommended Course Textbook

Linear Algebra with Applications, 4th Edition
by Otto Bretscher

ISBN: 0136-00926-3
ISBN 13: 978-0136-00926-9
Publisher: Prentice Hall

Bretscher, 4th Edition

Compare prices:
Linear Algebra With
Applications, 4th Ed.


Linear Algebra with Applications, 3rd Edition
by Otto Bretscher

ISBN: 0131-45334-3
ISBN 13: 978-0131-45334-0
Publisher: Prentice Hall
Bretscher, 3rd Edition

Compare prices:
Linear Algebra With
Applications, 3rd Ed.


Linear Algebra with Applications, 5th Edition
by Otto Bretscher

ISBN: 0321-79697-7
ISBN 13: 978-0321-79697-4
Publisher: Prentice Hall
Bretscher, 5th Edition

Compare prices:
Linear Algebra With
Applications, 5th Ed.

The older 2nd edition of the text may also be used. The material is fundamentally the same in all editions and all homework assignments will be made available as printable PDFs. Additional supplements on various topics in differential equations will also be made available during the course.

A key matching HW exercises in different editions is available on request.

A copy of the text will be put on reserve in the Grossman Library in Sever Hall.

Go for a Walk

AMC Local Walks:



Thanks for attending the course!

Course grades will be available online Tuesday, May 28.

Approximate letter grades for Exam #1
Total points on exam was 50. Median score was 42.0.
Mean score was 38.7. Standard deviation was 9.7.
Scores are posted on the Canvas site.
score grade   score grade
45+ A 31+ C+
43+ A– 28+ C
40+ B+ 25+ C–
37+ B 23+ D
34+ B– 0-22 E
Approximate letter grades for Exam #2
Total points on exam was 50. Median score was 46.0.
Mean score was 43.8. Standard deviation was 7.1.
Scores are posted on the Canvas site.
score grade   score grade
46+ A 33+ C+
44+ A– 30+ C
41+ B+ 27+ C–
39+ B 25+ D
36+ B– 0-24 E
Approximate letter grades for Final Exam
Total points on exam was 80. Median score was 58.
Mean score was 55.4. Standard deviation was 15.0.
Scores are posted on the Canvas site.
score grade   score grade
70+ A 48+ C+
64+ A– 44+ C
60+ B+ 40+ C–
56+ B 36+ D
52+ B– 0-35 E

Course grades will be available online Tuesday, May 28.

Here's a website that has a good java-based tool for showing vector fields and flows: http://math.rice.edu/~dfield/dfpp.html. The current version requires you to download a Java executable file to your own computer and to run it locally on your own machine. You can customize various options. You can also print the graphs. Choose the PPlane option.

Each week on Thursday evening before class (starting February 7) there will be an optional pre-class Q&A session starting at 7:00pm (or soon thereafter) - in Harvard Hall 103. The primary purpose of this Q&A session will be to take last-minute questions on the assigned homework problems after you have struggled with them. Robert Winters will conduct the session.

Renee Chipman's online discussion section meets Mondays (starting Feb 4) from 2:00pm to 3:00pm plus an additional half-hour at the end if there are any lingering questions. This session should be accessible via the Canvas site - most likely under Section Meetings (or something similar). You may be asked to download and install an executable file in order to participate.

Jeremy Marcq will also be conducting an online section on Fridays, 1:00pm to 2:00pm. The link is accessible via the Canvas site.

Needless to say, these section meetings are purely optional and you can participate in either of them.

Math E-21b Course Information and Syllabus (Spring 2019)       PDF version (Spring 2019)
These may still be amended slightly, but the essential details should be the same.

Classes will meet every Thursday evening from 8:00pm to 10:00pm in Harvard Hall 201 (next to Johnson Gate) starting January 31. There will also be optional online weekly problem sessions (conducted by TAs Jeremy Marcq and Renée Chipman) at days and times and locations to be determined. TA Renée Chipman will also meet with individuals and small groups on request. Students are not assigned to sections - any student may attend (online) the sessions by either Jeremy, Renée, or both.

Each week on Thursday evening before class (starting February 7) there will also be an optional pre-class Q&A session starting at 7:00pm (or soon thereafter) - in Harvard Hall 103. The primary purpose of this Q&A session will be to take last-minute questions on the assigned homework problems after you have struggled with them. Robert Winters will conduct the session.

Important Dates - Harvard University Extension School - Spring 2019

Course drop deadline for half-tuition refund   Sun, Feb 10, 2019
Withdrawal for WD/WN grade deadline (no refund)     Fri, Apr 26, 2019
Final Exam (Math E-21b) Thurs, May 16, 2019
Grades available online Tues, May 28, 2019

Birds on a wire

Questions, questions, questions..... and some answers

1) I usually post announcements and assignments initially on the working course website: http://math.rwinters.com/E21b/
I also then post them on the Canvas Site which is linked from there, but it's best to first check on this (non-Canvas) site.

2) All of the lectures will be live-streamed and recorded and made available within about one day. All of the lectures will be available via a link on the course's Canvas Site for the duration of the course and for a few weeks thereafter.

3) The main materials of the course are the lectures, the weekly Lecture Notes, the Bretscher text (best to get an inexpensive one online - links for best deals on the course website), and, of course, the homework assignments.

4) People who attend class in person can submit HW on paper in class. Distance students will be able to submit HW as a single scanned PDF (easily readable and with a reasonable file size, i.e. not scanned at an unnecessarily high resolution). Since classes are on Thursday and the recorded lectures are available on Friday (usually), I generally have assignments due no later than the following Saturday night. All students have the option to submit their assignments online, including those who regularly attend the lectures.

5) We will have two midterm exams and a final exam. The Calendar is given at: http://math.rwinters.com/E21b/calendar.htm. Most students who can take exams in person will do so on the scheduled date of the exam. Distance students are responsible for finding a qualified proctor near their home who will administer and monitor them throughout the duration of their exam. The proctor must be approved by the Distance Exams Office at least a week prior to the exam. Proctors scan and send the completed exams to the Exams Office, and they then make them available to the Instructor for grading. Students can find more information at the Extension School website here: https://www.extension.harvard.edu/exams-online-courses

6) Since Linear Algebra is largely independent of the Calculus sequence, any prerequisites other than general mathematical competence are really just recommendations or suggestions. Even the Placement Test is really for your own use to see if you're ready for the course. It is most certainly not required for registration.

7) The "Graduate" credit option is primarily for students enrolled in certain Extension School graduate programs such as the "Math for Teaching" program. All other students (including high school students) should register for the "Undergraduate" option or the Noncredit option (if you will not be submitting homework or taking exams).

The Harvard Extension School is committed to providing an accessible academic community. The Accessibility Office offers a variety of accommodations and services to students with documented disabilities.

Please visit https://www.extension.harvard.edu/resources-policies/resources/disability-services-accessibility for more information.

We prefer that assignments be submitted on paper for those who attend class in person. Other students should submit their assignments as a single scanned PDF file, but this option is also open to local students for any given assignment.

A basic standard is that you should not scan at resolution greater than about 120dpi. Anything beyond that is not necessary for handwritten work and dramatically increases file size. Black & White scans are preferred unless there is a good reason to scan using color. Please also make sure that the contrast is adjusted properly so that all work is clearly legible.

Photographs of assignments will not be accepted - scans only.

Based on class size and practical limits, we will select a subset of each assignment for grading. Solutions to all problems will be posted after the due date.

Homework deadlines: All homework is due in class the week after it is assigned - either a) in class or b) online via the Canvas site - typically by 11:59pm Eastern Time on the Saturday after the following lecture. Late assignments will generally not be accepted unless cleared in advance with either one of the Teaching Assistants (who will be doing the grading) or the Instructor. Your lowest homework grade will not factor into your course grade. No credit will be given after solutions have been posted.

Homework policies: Please write up your solutions neatly and show all work. If you use a calculator or other technology as part of your solution, this should be clearly indicated. When asked to "prove" or "justify" a statement, you must write clear verbal explanations in addition to doing out the math; the idea behind a proof is to convince the reader that the statement is true. Most problems will be marked out of 5 points (with the exception of some longer problems, which may be 10 points).

Guidelines: Please 1) staple all your pages; 2) write the assignment number on the top; and 3) include your name.

Completeness: It is always best to attempt every problem and to turn in each homework assignment even if there are some errors or omissions. Please always include work you have done that led to the final solution; this way we can point out where you have made an error. However, please don't include pages and pages of extraneous work. Your work will be graded on quality and not by weight or the number of pages submitted.

Collecting and Returning: We will collect homework submitted on paper each week at the lecture. Graded homework will usually be available before and after class. Homework assignments submitted online will generally be graded with only minimal comment, so you should consult the posted solutions to better understand whatever errors you may have made.

If there are any questions on these policies, please let us know. We look forward to working with you all!

Homework: Total points and median scores

HW #1 #2 #3 #4 #5 #6 #7 #8 #9 #10 #11 #12 #13
Total Points 50 50 50 50 50 50 50 50 50 50 50 50 50
Median Score 47 48 44 46 45 48 45 47 45 46 46 46 44
Mean Score 42.8 42.8 38.9 40.6 38.8 41.0 34.6 38.5 36.2 36.6 36.1 37.8 37.8

Note: The mean on the HW scores is skewed downward due to some students failing to submit HW.

Academic Integrity Letter from Robert Neugeboren, Dean of Students, Harvard Extension School (2014)
In particular: All work submitted ... must be your own words and ideas, or the source must be clearly acknowledged. Copying material from public websites (or other sources) ... is plagiarism, unless the source is properly cited ....

Harvard Hall - This is where class meets (from a 1910 postcard - 100 years ago).

Restore America's competitiveness!
Don't forget to pick up a textbook!

A letter to the New England Courant, dated May 14, 1722, and actually written by Benjamin Franklin under the
pseudonym "Silence Dogood." This was one of 14 letters by Silence Dogood and concerns Harvard University.

with the permission of Bill Griffith

posted with the permission of Bill Griffith

A word about calculators:

Though you can do this course without a matrix-capable calculator or mathematical software, it's certainly easier if you have an electronic servant to handle the drudge work. I use a TI-85 (no longer sold, but a good buy if you can find a used one) and I've been very happy with it. Ideally, you'll want a calculator that can find the reduced row echelon form of a matrix (RREF). You might also want one that can calculate determinants, eigenvalues, and eigenvectors, but that's a lesser priority. The TI-83 Plus, the TI-84 Plus, the TI-84 Plus Silver Edition, the TI-86 (also discontinued), and the TI-89 calculators can handle these operations. You don't need anything fancier than this.

One feature that I find very handy is the ability to display fractions and convert a decimal expression (for a rational number) to a fraction. That's useful when translating the results of an RREF calculation into parametric equations for a solution to a system of linear equations.

Here's a link that gives a comparison of the various TI calculators. Other manufactures also produce calculators that will work well with this course.

Linear Algebra Toolkit - an excellent online collection of tools that will not only do the calculations but also walk you through the steps. (The PERL scripts are written by Przemyslaw Bogacki.)

Useful Links:

Download your free Adobe Acrobat Reader for reading and printing PDF formatted documents.

Please send comments to Robert Winters.

URL: http://math.rwinters.com/E21b

Last modified: Monday, May 27, 2019 11:50 AM