Harvard Summer School 
Instructor: Course Assistant: 
If you're not sure whether the contents of this page (or any other page) are current, Course Textbook Linear Algebra with Applications, 4th Edition
OR Linear Algebra with Applications, 3rd Edition
OR Linear Algebra with Applications, 5th Edition
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: Thanks for attending the course! Course grades have been submitted to the Summer School. Grades will be available online via the Summer School on Wednesday, August 19.  RW
Course Calendar (2015) Math S21b syllabus (2015  HTML) Printable syllabus (2015  PDF) We will meet every Monday, Wednesday, and Friday (except Fri, July 3) from 9:30am to 11:30am in Harvard Hall 202. Robert Winters' Office Hours: Mondays, Wednesdays, and Fridays following class either in our classroom or in my office (SC 127). Other times may be scheduled by appointment. On some days I may go to lunch after class and return to the office afterwards. Jeremy Marcq will have his weekly section on Wednesdays from 12:302:00pm in Sever Hall 106. Jeremy's office hours will be on Mondays from 12pm to 3pm in Sever Hall 106. Topics to be covered include GaussJordan 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 leastsquares approximation; inner product spaces; determinants; eigenvalues, eigenvectors, and the spectral theorem; discrete and continuous dynamical systems; phaseplane 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 PrenticeHall. 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. Note: If you must turn in homework assignments outside of class, you can use the Math S21b 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 righthand wall.
Week #1 Notes: Lecture #1 Notes Lecture #2 Notes Lecture #3 Notes Week #2 Notes: Lecture #4 Notes Lecture #5 Notes Lecture #67 Notes Week #3 Notes: Lecture #78 Notes Lecture #9 Notes Week #4 Notes: Lecture #10 Notes Lecture #11 Notes Lecture #12 Notes Week #5 Notes: Lecture #13 Notes Lecture #1415 Notes Week #6 Notes: Lecture #16 Notes Supplement on nonlinear systems
by permission of Bill Griffith
Useful Links:
Download your free Adobe Acrobat Reader for reading and printing PDF formatted documents. Please send comments to Robert Winters. 