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2010_psets

# 2010_psets - 18.02 HOMEWORK#1 DUE BJORN POONEN Page numbers...

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18.02 HOMEWORK #1, DUE SEPTEMBER 16, 2010 BJORN POONEN Page numbers refer to Edwards & Penney. Problem numbers like 1A-1 refer to the Sup- plementary Notes. “(After Sept. 10)” means that the math needed for the problem should have been covered in lecture by that date. Make sure that you know how to do all exercises listed, but hand in only those labelled with point values . 1. Part A (After Sept. 9) p. 777: 21, 31 (After Sept. 9) p. 786: 13* (for the vectors in 1 only) (5 pts.), 39, 58* (10 pts.), 70* (5 pts.) (After Sept. 9) 1A-1b, 1A-5, 1A-8, 1A-12* (10 pts.) (After Sept. 9) 1B-1b, 1B-3b, 1B-6 (in c, change “empty” to “a point”), 1B-8, 1B-12 (After Sept. 9) 1C-2b, 1C-3b, 1C-7, 1C-9 (After Sept. 10) 1D-1a, 1D-3, 1D-5 (After Sept. 10) 1F-5a, 1F-8a, 1F-11b* (10 pts.) (After Sept. 14) 1G-2a, 1G-3, 1G-4 (After Sept. 14) 1H-3(a,c), 1H-6* (10 pts.), 1H-8* (10 pts.) 2. Part B B.1) (After Sept. 10, 15 pts.) A nilpotent matrix is a square matrix A such that A n = 0 for some positive integer n . (a) Show that ( 0 1 0 0 ) is nilpotent. (b) Show that if A is a nilpotent m × m matrix for some m 1, then det A = 0. (c) If A and B are nilpotent matrices of the same size, must A + B be nilpotent? (If YES, explain why. If NO, exhibit a pair of specific nilpotent matrices A and B such that A + B is not nilpotent.) B.2) (After Sept. 10, 15 pts.) Let A be the 2 × 2 matrix such that for each v R 2 , the vector A v is the mirror reflection of v in the x -axis. Let B be the 2 × 2 matrix such that for each v R 2 , the vector B v is the mirror reflection of v in the line y = x . (a) Find the matrices A and B . (b) Compute AB . (c) What transformation of the plane does AB give? (In other words, given v R 2 , what is the geometric relationship of ( AB ) v to v ? If it’s unclear, try a few test inputs.) B.3) (After Sept. 10, 10 pts.) On the exterior of a (not necessarily regular) tetrahedron, one vector is erected perpendicularly to each face, pointing outwards, and its length is equal to the area of the face. Show that the sum of these four vectors is 0 . Reminder: Please write “Sources consulted: none” at the top of your homework, or list your (animate and inanimate) sources. See the course information sheet for details. 1

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18.02 HOMEWORK #2, DUE SEPTEMBER 23, 2010 BJORN POONEN 1. Part A (After Sept. 16) 1E-1, 1E-2, 1E-3, 1E-6, 1E-7* (15 pts.) (assume that the lines are not parallel) (After Sept. 16) p. 650: 34* (15 pts.) (After Sept. 16) p. 802: 37, 39, 57* (10 pts.) (After Sept. 16) 1I-2, 1I-3, 1I-5 (After Sept. 17) 1J-2, 1J-5, 1J-6 (After Sept. 21) p. 858: 27, 37, 53–58 (ignore the labels on the axes in Figures 13.2.39–44, since they are misleading or wrong, depending on your interpretation) (After Sept. 21) 2A-1, 2A-5a (After Sept. 21) 2B-1a, 2B-3, 2B-10* (10 pts.) 2. Part B B.1) (After Sept. 17) Let PQ be the major axis of the elliptical orbit of a planet orbiting a sun. Suppose that the planet is moving at 80 km/s when it is at P , and 20 km/s when it is at Q . (a) (10 pts.) How far is the sun from P ? (Express the answer as a fraction of PQ .) (b) (15 pts.) How fast is the planet moving when it is at an end of the minor axis?
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2010_psets - 18.02 HOMEWORK#1 DUE BJORN POONEN Page numbers...

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