Summer 2017

Harvard University Summer School
Mathematics S-21b Summer 2017
Linear Algebra
and Differential Equations

Instructor:
Robert Winters

Course Assistant:
Jeremy Marcq

Calendar of topics and
homework assignments
Solutions
(password required)
Canvas site

If you're not sure whether the contents of this page (or any other page) are current,
hit F5 to refresh the page!


Course Textbook

Linear Algebra with Applications, 4th Edition
by Otto Bretscher

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

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

OR

Linear Algebra with Applications, 3rd Edition
by Otto Bretscher

ISBN: 0131-45334-3
ISBN 13: 978-0131-45334-0
Publisher: Prentice Hall

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

OR

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.

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

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.

 

Announcements:

Exam #1 took place during the latter part of class on Wed, July 5. It covered topics in Chapter 1-3 of the text.

Exam #2 took place during the latter part of class on Wed, July 19. It covered topics in Chapter 4-6 and Chapter 7, Sections 7.1-7.4 of the text. The graded exams will be returned in class on Monday. Scores are accessible via the Canvas site.

The Final Exam took place on Friday, August 4 from 8:30am to 11:30am in Harvard Hall 201.

Approximate letter grades for Exam #1
Total points on exam was 50. Median score was 43 (86%).
Mean score was 38.3 (76.6%). Standard deviation was 10.0.
score grade    score grade
45+ A 30+ C+
42+ A- 27+ C
39+ B+ 24+ C-
36+ B 21+ D
33+ B- 0-20 E
Practice Exam #1    Solutions
Exam #1 solutions
    
Approximate letter grades for Exam #2
Total points on exam was 50. Median score was 40.5 (81%).
Mean score was 38.9 (77.8%). Standard deviation was 8.6.
score grade    score grade
45+ A 30+ C+
42+ A- 27+ C
39+ B+ 24+ C-
36+ B 22+ D
33+ B- 0-21 E
Practice Exam #2     Solutions
Exam #2 solutions
Approximate letter grades for Final Exam
Total points on exam was 86. Median score was 64 (74.4%).
Mean score was 60.7 (70.6%). Standard deviation was 17.7.
score grade    score grade
78+ A 51+ C+
72+ A- 47+ C
66+ B+ 43+ C-
60+ B 39+ D
55+ B- 0-38 E
Practice Final Exam     Solutions

Note regarding submission of homework via email: We REALLY prefer that you do not submit homework as email attachments. Paper is so much easier to work with for the grading of homework. However, if you feel you must turn in an assignment via email, here are a few rules: (1) Get the OK first from Jeremy; (2) please don't send photographs unless the images are as good as what might be expected from a scanner; and (3) limit your file size to no more than about 150KB per page. If you don't know how to limit the file size, then don't submit your homework in this manner.

We REALLY would prefer paper.


HW #10 problems (PDF) - not to be turned in, but do them. The topics will be covered on the Final Exam.

HW #9 problems (PDF) - due Mon, July 31. Turn in only those problems designated as such. There are also additional practice problems that you may try at your convenience but which are not to be turned in.

HW #8 problems (PDF) - due Wed, July 26. Turn in only those problems designated as such. There are also additional practice problems that you may try at your convenience but which are not to be turned in.

HW #7 problems (PDF) - due Fri, July 21. Turn in only those problems designated as such. There are also additional practice problems that you may try at your convenience but which are not to be turned in.


We meet every Monday, Wednesday, and Friday from 9:30am to 11:30am in Harvard Hall 201.

Robert Winters' Office Hours: Mondays, Wednesdays, and Fridays immediately following class (if necessary we'll move from Harvard 201 elsewhere to continue the conversation). Other times may be scheduled by appointment, if necessary. My Summer School office is Science Center 308.

TA Jeremy Marcq has scheduled his weekly section each Friday from 12:00pm to 1:00pm at a location to be determined. This day/time is preliminary and may change. Jeremy will communicate this information via email.


Course Calendar (2017)      Math S21b syllabus (2017 - HTML)      Printable syllabus (2017 - PDF)


Topics to be covered include Gauss-Jordan reduction and systems of linear equations; matrices and linear transformations; linear independence; subspaces; matrices and coordinates relative to different bases; general linear spaces; orthogonality and least-squares approximation; inner product spaces; determinants; eigenvalues, eigenvectors, and the spectral theorem; discrete and continuous dynamical systems; phase-plane analysis of linear and nonlinear systems of ordinary differential equations; and function spaces and differential operators.

This course is primarily a course in Linear Algebra. It has always included applications to Differential Equations (perhaps 15% of the course), but the intention has never been to serve as a substitute for a full course in Differential Equations. This summer we will likely provide more focus on the applications of Linear Algebra to Differential Equations than in summers past.

Text: Linear Algebra With Applications, 4th Edition (2008) or 3rd Edition (2005) by Otto Bretscher, published by Prentice-Hall. The new 5th Edition (2013) of the text may also be used, but HW problems are currently keyed to the 4th Edition. Use the links to the left to find the best online prices for a text (used copies and older editions are considerably less expensive). We will cover almost all topics in this book, and homework will be assigned from its large collection of exercises. The material is fundamentally the same in all editions and all homework assignments will be made available as printable PDFs. A key matching HW exercises in different editions is available on request. Additional supplements on various topics in differential equations will also be made available during the course.

Harvard Hall


Answers to frequently asked questions:

There is relatively little overlap between the Multivariable Calculus course and the Linear Algebra/Differential Equations courses, and they can be taken in either order or simultaneously. That said, there are a few vector-specific items that are necessary for the Math S-21b course but these are things that you may already know or can pick up pretty quickly. These include vector basics (adding and scaling) and the dot product (which we definitely will briefly develop and review). For the material on Differential Equations that we will primarily cover late in the course, it is not necessary to have had any previous exposure to Differential Equations.


The Math S-21b course is primarily a course in Linear Algebra with several applications to Differential Equations. It does not serve as a substitute for a full course in Differential Equations.


Harvard's Multivariable Calculus (21a) and Linear Algebra (21b) courses can be taken in either order or concurrently. During the Summer School we always have a number of students taking both courses, but it really is a lot of work to do that given the rapid pace of the Summer School courses.


Regarding any questions about being sufficiently "in-depth", we prove every result used in the course. It's not a course in which facts are simply given and applied. It's also not a course in "the art of proof" in that we tend to focus on the aspects of Linear Algebra that arise in many other contexts - applied and otherwise.


Note: If you must turn in homework assignments outside of class, you can use the Math S-21b Dropbox located on the 2nd floor of the Science Center. As you exit the elevators on the 2nd floor, turn left and take an immediate left down the corridor with wooden mailboxes mounted on the right-hand wall.

HW Assignment #1 #2 #3 #4 #5 #6 #7 #8 #9
Total Pts. 135 130 120 70 130 100 120 110 90
median score 132 123 116 68 126 94 -- -- --

Week #1 Notes: Lecture #1 Notes     Lecture #2 Notes     Lecture #3 Notes

Week #2 Notes: Lecture #4 Notes     Lecture #5 Notes     Lecture #6-7 Notes

Week #3 Notes: Lecture #7-8 Notes     Lecture #9 Notes

Week #4 Notes: Lecture #10 Notes     Lecture #11 Notes     Lecture #12 Notes

Week #5 Notes: Lecture #13 Notes     Lecture #14-15 Notes

Week #6 Notes: Lecture #16 Notes      Supplement on nonlinear systems

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

Birds on a wire

Out of the Equation - Zippy the Pinhead, by Bill Griffith

by permission of Bill Griffith



(adapted from a Gary Larsen cartoon)



Click for today's comic.

Useful Links:


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

Please send comments to Robert Winters.
URL: math.rwinters.com/S21b
Last modified: Saturday, August 5, 2017 6:51 PM