Harvard Extension School
Mathematics E-21b - Spring 2019
Recommended Course Textbook
Linear Algebra with Applications, 4th Edition
Linear Algebra with Applications, 3rd Edition
Linear Algebra with Applications, 5th Edition
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.
AMC Local Walks:
Thursday, May 16 – Two-hour FINAL EXAM from 8:00pm to 10:00pm in Emerson 105
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. 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
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.
HW #13 problems (PDF) - keyed to 4th Edition (due in class on Thurs, May 9 or online by Sat, May 11, 11:59pm EDT)
Sampler of Phase Plane diagrams for uncoupled, coupled, period, and stable sink linear systems and a nonlinear system
Exam #2 took place on Thurs, Apr 25 during the latter part of class. The exam covered topics from Chapers 4-7 of the text (Lectures #5-11).
Exam #1 took place on Thurs, March 7 during the latter half of class. Calculators were permitted, but that's it.
Topics from Chapters 1-4 of the text, i.e. solving simultaneously m equations in n unknowns; row reduction and row operations, reduced row echelon form (RREF), parameterization of solutions; vector and matrix forms of systems of linear equations; linear transformations from Rm to Rn defined by matrices; inverse of a matrix, matrix algebra; subspaces, image and kernel of a linear transformation; span, linear independence; basis and dimension of a subspace; Rank-Nullity Theorem; coordinates of a vector relative to a basis; matrix of a linear transformation relative to an alternate basis; vector spaces and their subspaces; general linear transformations and their matrices relative to a basis or bases, isomorphisms.
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.
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
Questions, questions, questions..... and some answers
1) I usually post announcements and assignments initially on the working course website: http://math.rwinters.com/E21b/
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: Total points and median scores
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)
A letter to the New England Courant, dated May 14, 1722, and actually written by Benjamin Franklin under the
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.)
Download your free Adobe Acrobat Reader for reading and printing PDF formatted documents.
Please send comments to Robert Winters.
Last modified: Monday, May 20, 2019 9:59 PM