HW #10 problems (PDF) - do them, don’t turn them in. The solutions are posted for reference.

Supplement on nonlinear systems     Solutions

PPLANE - java-based tool for showing vector fields and flows: https://www.cs.unm.edu/~joel/dfield/ 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. [How Java is called varies on computer and operating system, so we may have to provide some additional documentation.] [Also available here]

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HW #9 problems (PDF - keyed to 4th Edition) - due in Canvas no later than Mon, Aug 5, 11:59pm

Lecture #16-17 Notes     Lecture #18 Notes (we’ll be covering a relatively small part of these - mainly a reference)

Supplement on Matrix Methods for Solving Systems of 1st Order Linear Differential Equations

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HW #8 problems (PDF - keyed to 4th Edition) - due in Canvas no later than Wed, July 31, 11:59pm

Lecture #14 Notes     Lecture #15 Notes

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HW #7 problems (PDF - keyed to 4th Edition) - due in Canvas no later than Fri, July 26, 11:59pm

Lecture #12 Notes     Lecture #13 Notes (updated July 21)

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Exam #2 will take place beginning on Fri, July 26 using Proctorio within Canvas. There will be a 24-hour window starting at 12:00pm (noon) on Fri, July 26 and ending at 12:00pm (noon) on Sat, July 27 during which you will be able to take the exam, scan your work, and submit the completed exam - just as we did for the first exam. Please do not wait until the last minute to do the exam and make sure you are fully familiar with the protocol for taking an exam in Proctorio. The exam will cover topics from Chapter 4 through Chapter 7, sections 7.1-7.4, i.e. through topics we cover on Monday (with a few added details on Wednesday).

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HW #6 problems (PDF - keyed to 4th Edition) - due in Canvas no later than Mon, July 22, 11:59pm

Lecture #10 Notes     Lecture #11 Notes

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HW #5 problems (PDF - keyed to 4th Edition) - due in Canvas no later than Wed, July 17, 11:59pm

Lecture #7-8 Notes     Lecture #9 Notes     Supplement on Least Squares in Economics

A caution in applying the Method of Least Squares in regression analysis – Outliers

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HW #4 problems (PDF - keyed to 4th Edition) - due in Canvas no later than Fri, July 12, 11:59pm

Lecture #6 Notes     Supplement on Representation of Functions in Different Coordinates

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

MWF 9:30am-11:30am in Harvard Hall 101, June 24 to August 2 (Aug 5 - Aug 9 for review and Final Exam)

Participation Options: On Campus, Live Stream, On Demand Online
Access the Live Stream lectures and recitations in Canvas under “Zoom”. Recordings will appear later in Canvas under “Class Recordings”.

Lectures, section meetings, office hours - all times EDT - schedule subject to change
Sun	Mon	Tues	Wed	Thurs	Fri	Sat
---	Lecture: 9:30am-11:30am (Zoom Chat - Renée) RW Office Hour (Zoom): to be determined Renée Recitation (Zoom): 1:30-3:30pm	Jeremy Recitation (Zoom): 9:00am-10:30am Jeremy Recitation (Zoom): 6:00-7:30pm	Lecture: 9:30am-11:30am (Zoom Chat - Jeremy) Renée Recitation (Zoom): 1:30-3:30pm	Kris Office Hours (Zoom): 9:30-11:30am Kris Office Hours (Zoom): 6:30-8:30pm	Lecture: 9:30am-11:30am (Zoom Chat - Kris) RW Office Hour (Zoom): to be determined	---

Note: All recitations are optional and you may attend any of them.

This summer we’ll be meeting in Harvard Hall.

Lectures will meet every Monday, Wednesday, and Friday from 9:30am to 11:30am EDT in Harvard Hall 101 or via Live Stream or On Demand Online from Mon, June 24 through Fri, Aug 2. The week of Aug 5 - Aug 9 will be taken up with reviews and the Final Exam. There will be multiple (optional) problem sessions every week (mainly via Zoom) at times to be determined and subject to adjustment depending on demand.

Robert Winters will generally entertain questions informally after the Monday, Wednesday, and Friday lectures and also later via Zoom (see schedule above). Other times may be scheduled by appointment.

Students may attend at the scheduled meeting time either in person or via Live Stream or may watch recorded sessions on demand. The recorded sessions are available within 24 hours of the class meeting time (and usually much sooner).

In addition to the recitations condicted by the Teaching Assistants and if you feel the need, Academic Tutoring Services is also available for Math S-21B students this summer. The registration process is simple. Students can find detailed instructions on the OSA website on how to connect with a tutor who can assist them with their academic needs.

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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.

Note: 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.

For Harvard College students: Different from the versions of Math 21b taught in the fall and spring in the College, MATH S-21b does not count for the Quantitative Reasoning with Data requirement. However, it does count for the Science and Engineering distribution requirement.

Text: Linear Algebra With Applications, 4th Edition (2008) or 3rd Edition (2005) by Otto Bretscher, published by Prentice-Hall. The newer 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.

Answers to frequently asked questions:

Note: The “Graduate” credit option is primarily for students enrolled in certain graduate programs such as the “Math for Teaching” program in which it may fulfil certain requirements. 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).

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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.

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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.

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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.

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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.

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Note regarding submission of homework: We REALLY prefer that you do not submit homework as email attachments. Scanned submission as PDF files via Canvas will be the standard. However, if you feel you must turn in an assignment via email, here are a few rules: (1) Get the OK first from one of the Teaching Assistants; (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. [These same rules apply to assignments submitted via Canvas.]

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Regarding Homework: Your worked solutions should be submitted as a single scanned PDF file (not individual images or single-page PDFs). There are plenty of utilities available online to merge single images or single-page PDFs into one combined PDF file. We may establish limits on file size, but a basic standard is that you do not need to scan at resolution greater than about 150dpi. Anything beyond that is not necessary for handwritten work and can dramatically increase file size. Also, make sure your PDF page size is standard letter size (8½ x 11) or close to it.

By the way, my scanner actually scans at 300dpi but still manages to create a PDF with about 30kb per page file size, so it’s not just about the scanning resolution. It’s also about the file compression in the scanning software.

Also please make sure that the contrast is adjusted properly so that all work is clearly legible. As a general rule, your file size should be no more than about 150kb per page. Try experimenting with different settings or software to get ideal resolution and contrast. There is no reason to ever submit HW with file sizes 60 times what is necessary for great resolution and contrast.

We will generally select a subset of 10 problems from each assignment for grading. Solutions to all problems will be posted after the deadline.

Staying current with the assigned HW is essential.

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HW Assignment	#1	#2	#3	#4	#5	#6	#7	#8	#9
Total Pts.	50	50	50	50	50	50	50	50	50
median score	48	46	45	48	47.5	46	46	45	-
mean score	46.1	44.0	44.0	43.8	45.1	43.5	44.1	42.4	-

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Don’t forget to pick up a textbook!

by permission of Bill Griffith

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(adapted from a Gary Larsen cartoon)

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Click for today’s comic.

Useful Links:

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

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Please send comments to Robert Winters.
URL: math.rwinters.com/S21b
Last modified: Sunday, August 4, 2024 7:25 PM