Spring 2017

Harvard Extension School

Mathematics E-21b - Spring 2017

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

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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:



The two-hour Final Exam took place on Thursday, May 11 from 7:40pm to 9:40pm in Emerson Hall, Room 105.

Approximate letter grades for Exam #1
Total points on exam was 44. Median score was 34.0.
Mean score was 32.5. Standard deviation was 9.2.
score grade   score grade
40+ A 25+ C+
37+ A– 22+ C
34+ B+ 20+ C–
31+ B 18+ D
28+ B– 0-17 E
Approximate letter grades for Exam #2
Total points on exam was 54. Median score was 46.0.
Mean score was 43.7. Standard deviation was 8.9.
score grade   score grade
49+ A 34+ C+
46+ A– 31+ C
43+ B+ 28+ C–
40+ B 26+ D
37+ B– 0-25 E
Approximate letter grades for Final Exam
Total points on exam was 84. Median score was 61 (73%).
Mean score was 59 (70%). Standard deviation was 16.3.
score grade   score grade
76+ A 47+ C+
69+ A– 43+ C
61+ B+ 40+ C–
55+ B 36+ D
51+ B– 0-35 E

Exam #2 took place during the latter part of class on Thurs, April 20.

Exam #1 took place on Thurs, Mar 2 during the last hour of class.

Math E-21b Course Information and Syllabus (Spring 2017)       PDF version (Spring 2017)

Classes will meet every Thursday evening from 7:40pm to 9:40pm in Harvard Hall 201 (next to Johnson Gate) starting January 26. There will also be an optional weekly problem session (conducted by TA Jeremy Marcq) every Tuesday from 7:40pm to 8:40pm in Sever Hall 105. TA Renée Chipman will meet with individuals and small groups on request.

Each week on Thursday evening before class (starting February 2) there will also be an optional pre-class Q&A session starting at 6:40pm (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 2017

Registration ends Sun, Jan 22
Late registration (with $50 late fee) Mon, Jan 23 - Mon, Jan 30
Course and credit status changes deadline Mon, Jan 30
Course drop deadline for full-tuition refund Mon, Jan 30
Course drop deadline for half-tuition refund   Mon, Feb 6
Withdrawal for WD/WN grade deadline (no refund)     Fri, Apr 21
Final Exam Thurs, May 11
Grades available online Tues, May 23

Homework deadlines: All homework is due in class the week after it is assigned, or in the dropbox marked "Math E-21b" on the second floor of the Science Center by 9:40pm the day it is due so that it may be merged with all the other homework papers submitted in class. Assignments submitted by 8pm the following Sunday will receive a 5-point deduction (the equivalent of one problem), and assignments submitted after this will be worth half credit at most. You may turn in one assignment late, but you must notify one of the course assistants before the assignment's deadline to receive this "freebie". Additionally, 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. (In particular, using WolframAlpha is not acceptable, except to check your work.) 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 assignment # on the top; 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.

Collecting and Returning: We will collect your homework each week at the lecture, but homework may also be submitted via our mailbox on the 2nd Floor of the Science Center. Any provision for submitting homework via e-mail or otherwise should be worked out individually with the Teaching Assistants. Graded homework will usually be available before and after class.

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

Birds on a wire

Homework: Total points and median scores

HW #1 #2 #4 #5 #6 #7 #8 #9 #10 #11 #12 #13
Total Points 85 95 125 95 55 135 85 105 95 55 65 90
Median Score 80 91 118 91 52 127 82 103 93 52 60 86

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, May 14, 2017 2:28 PM