Math 331 Assignments

Spring Semester, 2002

Assignments are listed according to the date due.
Fri., May. 31: Follow-up Assignment (also available as PDF or DVI)
Fri., May. 17: Final Examination, 1:00 - 3:00
Wed., May. 8: Last Short Test (10 minutes)
Review Session: Bring Questions.
Mon., May. 6: Do the following:
  1. We have seen that each of the four classes of non-identity isometries of the plane is “closed” under conjugation by isometries.
    Does this remain true for each of the six classes of non-identity isometries of space?
  2. Can one determine in which of the six classes of non-identity isometries of space a given non-identity isometry lies on the basis of knowing (i) whether it preserves orientation and (ii) the dimension of its fixed point locus?
    If so, match the criteria with the classes. If not, what other similar information might be used to “separate” the classes?
Fri., May. 3: 389: 1 - 4
and answer the following questions:
  1. While aside from the identity there are four different kinds of isometries of the plane, how many different kinds of isometries of space are there aside from the identity? What are they?
  2. In the plane every isometry can be factored as the composition of at most 3 reflections. Is there a corresponding basic building block for the isometries of space? If so, how many might be needed to “factor” a given isometry?
  3. How many symmetries are admitted by
    1. a cube?
    2. a “regular” tetrahedron?
Wed., May. 1: Read § 11.6
368: 8
373: 6, 7, 9, 10
379: 4
Mon., Apr. 29: Read § 11.4 - 11.5
368: 1, 3, 4, 6
373: 1, 5, 6
Fri., Apr. 26: Read §§ 11.1 - 11.3
348: 3 - 7
360: 1 - 6
Wed., Apr. 24: Read § 10.7
Exercises (also available as PDF or DVI)
Mon., Apr. 22: Read §§ 10.4 - 10.5
Exercises (also available as PDF or DVI)
Fri., Apr. 19: Read §§ 10.1 - 10.3
318: 6
330: 5
338: 2
and the following exercises pertaining to isometries of the plane:
What results when the reflections in two intersecting lines are composed? Does the order of their composition matter?

What results when the reflections in two (different) parallel lines are composed? Does the order of their composition matter?

What type of affine transformation is obtained when a reflection is followed with the translation by a non-zero vector perpendicular to the axis of the reflection?
Wed., Apr. 17: Read § 9.6
209: 2, 3, 7, 9
212: 1
318: 3, 5
Mon., Apr. 15: Read § 6.8 - 6.9
306: 7
313: 7, 8
318: 4
196: 3, 4, 5
201: 2
Fri., Apr. 12: Read §§ 9.4 - 9.5
306: 3, 5
313: 2, 4, 5
318: 1
Wed., Apr. 10: Read §§ 9.1 - 9.3
303: 1, 3, 4
306: 1
and the following:
In the context of transformations of R^n, show that if the translation T_v by a vector v is conjugated by an arbitrary affine transformation f, the result is the translation T_w by a vector w, and, moreover, w is a linear function of v.
Mon., Apr. 8: Read §§ 6.6 - 6.7
189: 2
196: 6
201: 5
Fri., Apr. 5: Quiz
Read §§ 6.4 - 6.5
176: 6
185: 1 (use fig. 6.3), 2, 7
189: 1
Wed., Apr. 3: Read § 6.3
171: 2
175: 4, 5
180: 3, 5
Mon., Apr. 1: Read §§ 6.1 - 6.2
171: 1
175: 3
Mon., Mar. 25: Read § 4.7
118: 7, 10, 11
123: 1 - 4 and:
Do you find fault with the solution written below of the following?
Problem: Let A, B, and C be any points. If f is the unique translation for which f(A) = B, express f(C) as an affine combination of A, B, and C.
Response: Regard A, B, and C as an affine basis for the plane containing these points. Then in the corresponding barycentric coordinates A = (1, 0, 0), B = (0, 1, 0), and C = (0, 0, 1). Since f is a translation and f(A) = B, in barycentric coordinates f must be translation by the vector (0, 1, 0) - (1, 0, 0) = (-1, 1, 0). So f(C) must have barycentric coordinates (0, 0, 1) + (-1, 1, 0) = (-1, 1, 1). Therefore, f(C) = C + B - A.
Fri., Mar. 22: Quiz (a short re-test of midterm issues)
Read § 4.6
106: 6
118: 1, 3, 5
Wed., Mar. 20: Read § 4.4 - 4.5
106: 5
111: 1 - 4, 6
Mon., Mar. 18: Read § 4.1 - 4.3
102: 2, 4 - 6, 9 - 11
106: 1, 2
Fri., Mar. 15: Midterm Test
Wed., Mar. 13: Bring review questions
Mon., Mar. 11: Read § 8.6
285: 1, 3, 4
290: 4, 6, 7
Fri., Mar. 8: Read § 8.7
280: 4 - 6
290: 1, 2, 4
Wed., Mar. 6: Read § 8.4-8.5
275: 1 - 4
280: 1 - 3
Mon., Mar. 4: Review § 3.9
Read §§ 8.1-8.3
Problems 94: 2, 4
264: 2, 6
Extra Credit: Write up a full correct solution to the exercise on the point where the three angle bisectors of a triangle meet.
Fri., Feb. 22: 2nd Short Test
Read § 3.8
Problems 90: 5, 8, 9
94: 3
Wed., Feb. 20: Read § 3.7
Problems 84: 6, 7
90: 2, 3.
and the following:
Exercise: If A, B, and C are three non-collinear points, express the point where the angle bisectors of triangle ABC meet as a convex combination of A, B, and C.
Mon., Feb. 18: University Recess
Fri., Feb. 15: Read § 3.6
Problems 77: 6, 7
79: 1
84: 1-5
Wed., Feb. 13: Read §§ 3.1-3.5
Problems 62: 6-9
66: 1-6
77: 1-5
Mon., Feb. 11: First Short Test
Read § 2.7
Problems 62: 1 - 5
Fri., Feb. 8: Note: The announced first short test has been postponed to Monday because of the lateness of textbook re-stocking at the bookstore. The Department Office was told on Thursday afternoon (Feb. 7) that the store is now re-stocked.
Read § 2.6
Problems: 56: 1 - 4
Handout: Solutions to two past exercises (also available as PDF or DVI)
Wed., Feb. 6: Read §§ 2.1-2.4
Problems: 6: 1
39: 1, 6, 7, 9
44: 4
47: 6, 7, 8
Mon., Feb. 4: Read § 1.4:
Problems: 30: 3 - 7
Also: review a short list of web references on barycentric coordinates.
Fri., Feb. 1: Read § 1.3
Problems: 23: 3, 4
30: 1, 2
Mon., Jan. 28: Read § 1.8
Problems: 17: 1-6
23: 1,2
Fri., Jan. 25:Read §§ 1.5-1.6
Wed., Jan. 23:First Meeting: No Assignment

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