Spring 2020

Harvard Extension School

Mathematics E-21b - Spring 2020

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:



HW #3 problems (PDF) - keyed to 4th Edition (due Thurs, Feb 20 in class or Sat, Feb 22 online via Canvas)

Lecture #3 notes

HW #2 problems (PDF) - keyed to 4th Edition (due Thurs, Feb 13 in class or Sat, Feb 15 online via Canvas)

Lecture #2 notes

HW #1 problems (PDF) - keyed to 4th Edition (due Thurs, Feb 6 in class or Sat, Feb 8 online via Canvas)

Lecture #1 notes

[Multivariable Calculus Notes #1 for vector references]

Math E-21b Course Information and Syllabus (Spring 2020)       PDF version (Spring 2020)

Classes meet every Thursday evening from 8:00pm to 10:00pm in Maxwell-Dworkin G115 starting January 30, 2020. There will also be optional online weekly problem sessions (conducted by TAs Jeremy Marcq and Renée Chipman). 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.

Renee Chipman's online discussion section is scheduled to meet Mondays (starting Feb 3) 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's online discussion session will take place on Wednesdays (starting Feb 5) from 11:00am to noon. The link will be accessible via the Canvas site.

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

Each week on Thursday evening before class (starting February 13) there will be an optional pre-class Q&A session starting at 7:00pm (or soon thereafter) - currently in Northwest Science Building room B109 (though I hope this will change). 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.

For Thursday, February 6, I will endeavor to arrive early at Maxwell-Dworkin and be available informally for questions at the tables in the area outside our lecture hall. - RW

Important Dates - Harvard University Extension School - Spring 2020

Registration opens for degree candidates Thurs, Nov 7, 2019
Registration opens for all students Tues, Nov 12, 2019
Registration ends Thurs, Jan 23, 2020
Course change period - for registered students only Fri, Jan 24 to Sun, Feb 2, 2020
Classes begin (Math E-21b) Thurs, Jan 30, 2020
Course and credit status changes deadline Sun, Feb 2, 2020
Course drop deadline for full-tuition refund Sun, Feb 2, 2020
Course drop deadline for half-tuition refund   Sun, Feb 9, 2020
Withdrawal for WD/WN grade deadline (no refund)     Fri, Apr 24, 2020
Final Exam (Math E-21b) Thurs, May 14, 2020
Grades available online Tues, May 26, 2020

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 48.0                        
Mean Score 46.5                        

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: Sunday, February 16, 2020 9:15 PM