Spring 2025

Harvard Extension School

Mathematics E-21b - Spring 2025

Linear Algebra

Instructor:
Robert Winters

Course Assistants:
Jeremy Marcq
Renée Chipman
Kris Lokere

Handy Links:

Linear Algebra Toolkit

Math E-21b syllabus (2025)

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

OR

Linear Algebra with Applications, 3rd Edition
by Otto Bretscher
ISBN: 0131-45334-3
ISBN 13: 978-0131-45334-0
Publisher: Prentice Hall

OR

Linear Algebra with Applications, 5th Edition
by Otto Bretscher
ISBN: 0321-79697-7
ISBN 13: 978-0321-79697-4
Publisher: Prentice Hall

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 on reserve in the Grossman Library in Sever Hall.

Another excellent Linear Algebra text is Gilbert Strang’s “Introduction to Linear Algebra”, Sixth Edition.

ISBN: 978-17331466-7-8
Earlier editions of the Strang text
are also excellent references.

Announcements:

Math E-21b Course Information and Syllabus (Spring 2025)       PDF version (Spring 2025)

Course meetings: The class meets weekly on Thursdays, 8:00pm to 10:00pm [in person (Harvard 101) or via Zoom] or on-demand in Canvas beginning Thurs, Jan 30, 2023. Optional problem sessions conducted by our Teaching Assistants will be scheduled and are subject to change as the need arises. An optional Q&A session with the instructor may also be scheduled at a day and time to be determined.

Optional TA Section meetings (open to all, take place via Canvas/Zoom):
Kris Lokere: Tuesdays 7pm-8pm, Fridays 5pm-7pm - subject to change
Jeremy Marcq: Mondays 9am-10am, Wednesdays 6pm-7pm - subject to change
Renée Chipman: Monday and Thursday afternoons, 4:00pm-5:30pm - subject to change

These times may change or other times added as the need arises. The duration of meetings may be extended at the discretion of the TAs. Section meetings are accessed via Canvas/Zoom/Gather.

Prerequisites: Math E-16, or equivalent knowledge of algebra and calculus. You should be able to solve simple systems of equations and find the roots of polynomials. Also, you should be able to set up and solve simple differential equations. Math E-21a (or its equivalent) is not specifically necessary in order to take Math E-21b, but it will be very helpful if you have some familiarity with the algebra and geometry of lines and planes in R², R³, and possibly Rⁿ, and some basic knowledge about vector operations (especially scaling and adding vectors) and the dot product of two vectors.

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.)

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) 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 afterwards, I generally have assignments due no later than the following Saturday night. All students must submit their assignments online.

5) We will have two midterm exams and a final exam. Two midterm exams will take place online via Proctorio during 24-hour windows approx. on March 6-7 and April 24-25. There will be a two-hour final exam online via Proctorio on May 15. Your course grade will be computed according to the following scheme, subject to minor modification:

.25 (homework) + .40 (midterm exams) + .35 (final exam)

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 not required for registration.

7) The “Graduate” credit option is for students enrolled in or planning to be enrolled in the Extension School’s “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). Students registered for “Graduate” credit will be asked to complete additional work on supplemental topics.

8) Noncredit students will not be permitted to take the Final Exam. The Harvard Extension School will not permit students to change their registration from Noncredit to Undergraduate after the course is well underway, e.g. after any exams have occurred.

Course Credit Status

• To receive final grades and academic credit, students must register for undergraduate or graduate credit; this applies to all students regardless of whether they are pursuing HES degrees or certificates.
• To audit a course, students must register for the course for noncredit. No individual may audit a course without being officially enrolled. Noncredit students may not take quizzes, mid-terms or final exams, or submit final projects. We will, however, accept and grade HW assignments from noncredit students if they wish to submit assignments.
• Students requesting to change their credit status after the course change period must submit a Financial and Registration Appeal with documentation of an extenuating circumstance that prevented them from doing so during the course change period. Credit status changes are not eligible for an appeal after the course withdrawal deadline.

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.

Students should submit their assignments as a single scanned PDF file. 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.

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.

You are responsible for understanding Harvard Extension School policies on academic integrity (https://www.extension.harvard.edu/resources-policies/student-conduct/academic-integrity) and how to use sources responsibly. Not knowing the rules, misunderstanding the rules, running out of time, submitting the wrong draft, or being overwhelmed with multiple demands are not acceptable excuses. There are no excuses for failure to uphold academic integrity. To support your learning about academic citation rules, please visit the Harvard Extension School Tips to Avoid Plagiarism (https://www.extension.harvard.edu/resources-policies/resources/tips-avoid-plagiarism), where you’ll find links to the Harvard Guide to Using Sources and two free online 15-minute tutorials to test your knowledge of academic citation policy. The tutorials are anonymous open-learning tools.

In particular: “To avoid any suggestions of improper behavior during an exam, students should not communicate with other students during the exam. Neither should they refer to any books, papers, or use electronic devices during the exam without the permission of the instructor or proctor.”

“Breaches of academic integrity are subject to review by the Administrative Board and may be grounds for disciplinary action, up to and including requirement to withdraw from the Extension School and suspension of registration privileges.”

Important Dates - Harvard University Extension School - Spring 2025

Official Academic Calendar

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

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.

posted with the permission of Bill Griffith

Useful Links:

• Harvard Extension School
• Harvard University
• Harvard University Dept. of Mathematics

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: Thursday, February 6, 2025 2:11 PM