21 B
Mathematics Math21b Spring 2008
Linear Algebra and Differential Equations
Course Head: Oliver Knill
Office: SciCtr 434

The Final 21b Exam took place May 23 at 9:15 in Science Center B. The final exam schedule of the registrar. The date 23 was no accident!. The average of the final exam was 81 with standard deviation 14.
Practice exam I Practice exam II Practice exam III Practice exam IV Final exam
Solution practice exam 1 Solution practice exam 2 Solution practice exam 3 Solution practice exam 4 Solution final exam
Corrected Errata:
  • Practice exam 2, 2b, a constant C1 was missing in the solution. (Thanks to Peter Kim)
  • Practice exam 1, 14, a sign was wrong for the eigenvalue -n4+n2+1 of the operator T. (Thanks to Lisa Rotenstein and Charles Liu)
  • Practice exam 4, 13, the final solution had a factor 16 missing. There was also a sign error for the coefficients -cn2 = 1-16 n2 and a sin(n t) should have been sin(n x). (Thanks to Heidi Liu and Allen Yang)
  • Practice exam 3, 13, there was ghost factor 8 in the solution which should not have been there. (Thanks to Trung Nguyen.)
  • Practice exam 3, 11, in the intermediate step of the integration by parts, factors n were missing. The result is correct. (Thanks to Peter Kim)
  • Practice exam 3, TF 18, the solution labeled it false, but it is true. The explanation was correct. (Thanks to Taylor Yi).
  • I,li>Practice exam2, 13a), when rewriting the series with the bn, there had been a typo (4 instead of 8). (Thanks to Megan Blewett).
  • All students enrolled in this course are required without exception to attend the exam administered by the FAS Exams Office. Students not present at this exam will receive a grade of ABS for the course.
  • Instructors may not approve or administer a makeup exam to any student absent from the exam. Makeup exams may only be approved by the Administrative Board for undergraduates or the Deputy Registrar for graduate students, Special Students, or cross-registered students and will be administered by the FAS Exams Office during the scheduled makeup exam period.

Solution practice exam 1 Solution practice exam 2 Solution practice exam 3 Solution practice exam 4
Corrected Errata:
  • Practice exam 3, Problem 3: in the solution there had been a mixup with A and B. The matrix A has eigenvalues 10 and 4, the matrix B has eigenvalues 6 and 0. Thanks to Chenzi Xu. (Corrected April 5, 5 PM).
  • Practice exam 1, Problem 3 a). Typo: the eigenvalues are lambda,lambda2, lambda3 ... Thanks to William Weingarten (Corrected April 5, 12 PM).
  • Practice exam 4 2c), the eigenvalues are 1,2,3. Thanks to Allen Yang and Joe Stujenske (Corrected Apr 6 3 PM)
  • Practice exam 3, TF problem 5 solution, the formula for the reflection is 2P-1 not 1-2P. Thanks to Caitlin Crump, (corrected April 7 7 PM)
  • Practice exam 4, TF 2, typo in explanation. We have T(f) = f''-f which has eigenvalue -n2-1. (Thans to Chenzi Xu).

Photo by Karim Atiyeh, 3/4/2008
Photo by Karim Atiyeh, March 4, 2008, Harvard SCi Hall C, before start of exam
Solution practice exam 1 Solution practice exam 2 Solution practice exam 3
Corrected Errata:
  • Solution to Practice exam 2) problem 3a). Typo for the sign for y. It is positive 5. Thanks to Tzu-Ying Chuang. (corrected 3/1/2008 5 PM).
  • In the solution to Practice exam 1 problem 8). The rref(B) computation had not been finished yet. The zero row needed to be at the bottom. Thanks to Weike Wang. (corrected 3/2/2008 12:45 PM)
  • In the solution to Practice exam 3, problem 6) c) two signs were switched. The correct solution is {{0 0 2},{2 0 0},{0 -2 0}}. Thanks to Joe Stujenske (corrected 3/3/2008 10:30 PM)
  • In the solution to problem 7c), LaTeX typo: v_2- v-1 should have been v_2-v_1. Thanks to Jessica Villegas (corrected 3/3/2008, 11:00 PM).
  • In the solution to practice exam 2, TF 17, an equal sign was missing "=B2". Also, in the solution to practice exam 3, TF 18, one diagonal entry was wrong: A=diag(-1,-1,1) and B=diag(1,1,-1). Thanks to Jack Sun (corrected 3/4/2008 11 AM, rsp 4 PM).

Official policy at Havard is that "out of sequence midterm exams" are only offered for students observing religious holidays. We sometimes can accomodate 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.
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Math21b | Oliver Knill | Spring 2008 | Department of Mathematics | Faculty of Art and Sciences | Harvard University