21 B
Mathematics Math21b Spring 2009
Linear Algebra and Differential Equations
Course Head: Oliver Knill
Office: SciCtr 434

The Final exam schedule. Our final exam took place May 18th at 2:15 PM. The average score was 79/100 with a standard deviation of 12 (also the graph to the left is scaled to 100 as always). Here is the exam with the solutions: We had a global review in lecture Hall C from 3-4:30 on Tuesday May 12'th. Review slides [PDF] , For ipod [M4v] 100 Meg.
Here are 5 practice exams with solutions:
Practice exam I [PDF] Practice exam II [PDF] Practice exam III [PDF] Practice exam IV [PDF] Practice exam V [PDF]
Solution practice exam 1 Solution practice exam 2 Solution practice exam 3 Solution practice exam 4 Solution practice exam 5
Corrected typos and errors in solutions:

  • The 2. Midterm took place Tuesday 4/7, from 7-8:30pm in Hall C. The average score was 75. The standard deviation 13.
  • We had a global review on Sunday at 7-8:30 PM in Hall B. Here are the slides [PDF].
  • Sunday 4/5 3 PM: Practice I, Problem 6, AT b = [ 13,-1]T. Final result unchanged. Thanks to Jason Rosenfeld.
  • Monday 4/6 6:30 AM: Practice V, Problem 9b, mix-up of B and A in the answer to B. Thanks to Wilson Wu.
  • Monday 4/6 7 AM: Practice I, Problem TF 8, example is given for AB, not A+B. Thanks to Jim Danz
  • Monday 4/6 4 PM: Practice I, Problem 9, v3 had a typo in the sign. End result is correct. Thanks to Jim Danz
  • Tuesday 4/7 5 PM: Practice V,Problem 6, an entry in the matrix S was wrong. Thanks to Xuemei Zhai

The average score was 76 points. The standard deviation 10.
  • The 1. Midterm took place Tuesday 3/3, from 7-8:30pm in Hall C.
  • Review slides from review of March 1. [PDF],[M4V] Ipod 100 Meg.
  • The following practice exams should give you an idea about the structure of the first midterm. The first hourly covers the material from the first 4 weeks. mid
  • Midterm exam [PDF] ,Solutions
  • Practice Exam 1 [PDF] ,Solutions
  • Practice Exam 2 [PDF] ,Solutions
  • Practice Exam 3 [PDF] ,Solutions
  • Practice Exam 4 [PDF] ,Solutions
  • Typos and errors in the practice exam (corrected at the time indicated):
    • Monday 3/2 10 PM: Practice 1 TF16. The explanation had a confusing example. There is a better example now. Thanks to Jason Rosenfeld.
    • Monday 3/2 10 PM: Practice 2 4c) Remark in parenthesis: The matrix is a reflection about a line, not a point. Thanks to Jim Danz.
    • Monday 3/2 10 PM: Practice 2 7a) A minus 1 typo in the v part. End result is correct. Thanks to Jim Danz.
    • Tuesday 3/3 7 AM: Practice 1, B100 was used in b) instead of B. Thanks to Jon Noronha, Julie Xie.
    • Tuesday 3/3 7:30 AM: Practice 2, In the explanation, it should have been AS = SC, not AS = CS. Thanks to Jason Varnell.

    Official policy at Harvard is that "out of sequence midterm exams" are only offered for students observing religious holidays. We sometimes can accommodate students who happen to have an important sports event at the same time (it must be an actual game or tournament) or a concert or theater event (regular practice or rehearsals of course do not qualify to miss the exam). In case of a qualifying conflict, contact the course head to see whether it is possible to arrange the exam earlier at the same day. We will need a letter from your coach or director. The request has to come at least a week before the exam.
    Please send questions and comments to
    Math21b (Exam Group 1)| Oliver Knill | Spring 2009 | Department of Mathematics | Faculty of Art and Sciences | Harvard University