The explanations in the text are fairly detailed. So you are expected to study the text on your own, before and after the lectures. I will only have time to discuss the most essential aspects of the theory; you should read the explanations and examples to fill in details not carefully discussed in class. If you find places that need clarification, please raise these points to me either in or before class. This way, our discussion in the lecture time will be more focused.

The problems to be assigned are grouped into recommended and required. The recommended problems are scattered in the body of the text in each section, and have complete solutions in the back of the text, so will not be collected; the required problems are listed in the last section at the end of each chapter, and are collected for grading.

• Chapter I. Note that the word gradient in the text is the British English word for slope in American English. Please review the concepts of linear transformations, orthogonal matrices, eigenvalues and eigenvectors of matrices, and diagonalization of symmetric matrices. We will be needing these concepts mostly in the two or three dimensional setting.
  • Some aspects of Linear Algebra are used regularly in this course. Here are some review problems. Particularly relevant are problems 1, 4, 6, 7.
  • Sections 1.1, 1.2, due: Monday, Sept. 15.
    • Recommended: 1.1 : 3,6,7,8,9; 1.2: 2,4,7,8;
    • Required: 1.1: 4,5; 1.2: 3,5.
    • Here you will find the solutions to HW 1.
  • Sections 1.2, 1.3, due: Monday, Sept. 22.
    • Recommended: 1.3 : 1,2,3.
    • Required: 1.3: 1(a), (d); 1.2: Let AB be a diameter of a circle and P be a point on the same circle, other than A or B. Prove that AP is perpendicular to PB, and the angle PAB is equal to the angle between BP and the tangent to the circle at P.
    • Here you will find the solutions to HW 2.
• Chapter II. The most important geometric concepts of this chapter are Parallel Projections and Affine transformations. All geometric properties of affine transformations can be described by those of parallel projections, because Each parallel projection is an affine transformation (Theorem 5 in Section 2.2 ) and Any affine transformation can be expressed as the composite of two parallel projections (Theorem 6 of Section 2.2). The most important properties of affine transformations are the three listed on p. 57. With the Fundamental Theorem of Affine Geometry (on p.69), we can use an affine transformation to transform a given geometric problem into a more special one, where perhaps one can understand the situation more easily. All of Section 2.4 are about the application of this idea to geometric problems involving rectilinear figures. Section 2.5 contains applications of this idea to conic sections. For Section 2.5, we will only discuss Theorems 1-4 and the Corollary on p. 86 (and perhaps Theorem 7).
  • Sections 2.1, 2.2, due: Monday, Sept. 29.
    • Recommended: 2.1 3, 4, 5, 6, 7; 2.2 1, 2, 3.
    • Required: 2.1 2,5; 2.2 3, 4(a).
    • Here you will find the solutions to HW 3.
  • Sections 2.3, 2.4, due: Oct. 6
    • Recommended: 2.3: 2, 4, 5; 2.4: 1, 2, 4, 6.
    • Required: 2.3: 2(a), 4; 2.4: 4, 5.
    • Here you will find the solutions to HW 4.
    • Some of the completed homework do not contain enough explanations on the basic ideas and steps taken so that they are not easy or close to impossible to read. We also have a good portion of students who write fairly complete and readable solutions. I am posting here a copy of Nicole's completed homework for the second assignment---her work here is a good example of how to explain one's ideas and key steps; these are also good strategy for the students themselves, as they would help guide them through the multi-steps that are needed to complete the work. Note that the highlighted areas are comments that I added.
  • The first midterm is scheduled to be on Thursday Oct. 16. It will cover the material up to and including 3.2. Here are some practice problems. The following formula sheet will be made available on the exam. Hyun has brought to my attention a diagram error on p. 398 of the text: (i) the y' and y'' axes in the diagram should be x' and x'' axes, and the x' and x'' axes in the diagram should be the negative y' and y'' axes, as from the matrix P, the vector (x',y')=(1,0), which points along the x' axis, corresponds to the first column of P which is pointing toward the 4th quadrant, and the vector (x',y')=(0,1), which points along the y' axis, corresponds to the second column of P which is pointing toward the first quadrant; (ii) the major (longer) axis should be parallel to the corrected x' axis.
  • Here is the solution to the first midterm, and here is the distribution of scores on the first midterm: 91 87 86 83 77 69 69 68 65 65 63 63 62 58 53 51 50 49 49 47 43 38 26
  • Sections 2.4, 2.5, due: Oct. 13 (Some problems of 3.2 also due at the same date, see below).
    • Recommended: 2.4: 1, 2, 4, 6; 2.5: 1.
    • Required: 2.4: 6; 2.5: 1.
• Chapter III. The algebraic manipulations in the text becomes a bit more difficult, although the exercises do not involve much complicated algebra. The more difficult issue is perhaps the constant translation between a geometric concept (there are many new concepts in this chapter) and its algebraic representation. It is important not to get buried in the computations and forget about the geometry behind. Section 3.1, especially pp. 98--101, are worthy of repeated reading. In order not to be overcome by the algebra and abstract conception, it may be worth while to first read and understand the geometric meaning of the following theorems: Theorems 1 and 2 of 3.2 on p.106, and Theorem 3 of 3.2 on p.110. The really important geometric properties for us are refleted in Theorem 2 of 3.3 on p.119 and Theorem 4 of 3.3 on p.127. The only quantitative invariant of projective geometry is introduced through Theorems 3 and 4 of 3.5 on pp.140-141. I have writen up some notes on projective geometry. Hopefully they complement the discussion in the text. You are strongly adviced to read at least the Introductory Remarks in the notes before our first lecture on this chapter.
  • Section 3.2, due Oct. 13.
    • Recommended: 1, 2, 3, 4, 5, 6;
    • Required: 2(a), 3(a).
    • Here is the solution to this set of homework.
  • Sections 3.2 and 3.3, due Oct. 27.
    • Recommended: Section 3.2: 8,9,12; Section 3.3: 1, 2, 4, 5, 7.
    • Required: Section 3.2: 4(a), 5; Section 3.3: 3, 4.
  • Here is the solution to this set of homework, and
  • Sections 3.4 and 3.5, due Nov. 3 .
    For 3.5, we will only discuss the applications of Theorems 3 and 4 on pp.140-141 through their applications in 3.5.2, 3.5.3. Read 3.5.2, 3.5.3 on pp.143--146 carefully.
    • Recommended: Section 3.4:1, 2; Section 3.5: 4, 5, 6, 7, 8;
    • Required: Section 3.4: 2; Section 3.5: 4, 5; Also the following one: Suppose that A, B, C, and A', B', C' are collinear, respectively; and that BC' is parallel to B'C, that AC' is parallel to A'C. Prove that AB' is parallel to A'B. (Hint: You may use either the synthetic method or the coordinate method. If you use the synthetic method, be aware that the lines ABC and A'B'C' may intersect or be parallel, so you need to cover both cases; if you use the coordinate method, you may apply an affine transformation (would a projective transformation work?) to assume that C=(0,1), C'=(0,0), B'=(1,0), and A'=(a,0) for some a. You then need to set up an equation for the line ABC.)
    • Here is the solution to Homework 7.
• Chapter VII. We will cover 7.1, 7.2, 7.3.1, 7.3.3, and 7.3.4.
  • Sections 7.1, 7.2, Due Nov. 10.
    • Recommended: Section 7.1: 1, 2, 3, 4; Section 7.2: 1, 3, 4, 5, 8, 9.
    • Required: Section 7.1: 2, 3; Section 7.2: 1, 3.
  • Second midterm is scheduled on Thursday, Nov. 20 . It will cover chapters 3 and 7. Here is a set of practice problems, and here are solutions to the practice problems. Here is the solution key to the second midterm, and here is the distribution of scores for the second midterm: 97 97 95 94 92 92 90.5 90 85 85 83.5 82 80 80 79 78 77.5 74 73 71 71 68.
  • Here, together with additional solution, are the solution to Homework 8.
  • Section 7.3, Due Monday, Nov. 17
    • Recommended: 2, 5, 6, 7, 8;
    • Required: 2(a)-(b) (Hint: construct a median from one vertex and use Theorem 3 and one of the rules in Theorem 7 of 7.3.3), 3, 5.
    • Here are the solutions to this set of homework.
• Euclidean and non-Euclidean Geometry. In the remainder of the semester, our focus will be shifted to examining a few central issues in classical Euclidean Geometry and how the study of Euclid's Parallel Postulate led to the birth of a non-Euclidean geometry, called hyperbolic geometry today. I have written up some notes to help you put things into perspective. You should download these notes and study them before our discussions. Here are some notes along similar theme from last year. There is substantial overlap between these notes, though some discussion in the notes from last year have yet to incorporated into the current notes. For the lecture materials, we will be using Euclid's "Elements", Books I and III, and the following notes on non-Euclidean geometry that I prepared.
  • The following is an annotated list of Propositions in Books I and III of Euclid's "Elements" that I would like students to present in class. Please take a look at this list, pick your choice (and alternative, if case your top choice has been taken by others), and either email me your choice or sign up at our next meeting. Each student will be given up to 6 minutes for presentation, and an additional two minutes for discussions and answering questions. Your presentation should include a clear statement and explanation of the Proposition to be presented, a diagram to illustrate the geometric features of the Proposition, and a proof. If my annotation lists specific questions for the Proposition, then your presentation should give an answer to the questions. You don't have to follow Euclid's proof. You may also form small groups to work on a cluster of Propositions together, as your proof most likely will need to refer to neighboring Propositions. Also your groupmates can help you check the validity of your arguments.
    • Here is a list of propositions that have already been taken for presentation as of Nov. 24: I. 30, I. 33--38, III. 1--4, III. 10--15, III. 18, III. 19, III. 21, III. 32.
    • I. 30: Justify that the two given straight lines parallel to a given common straight line are in the same plane.
    • I. 33 -- 34: Properties of parallelograms. In what step(s) does Euclid's Postulate V (or its equivalent version) enter?
    • I. 35 -- 41: Areas of parallelograms and triangles. Here in what sense is the meaning of "equal" different from the one used in earlier Propositions? Do these propositions justify the area formula in Euclidean plane as one half times base times height? In what step(s) does Euclid's Postulate V (or its equivalent version)enter?
    • III. 1--4.
    • III. 10--13.
    • III. 14--15.
    • III. 16, 18, 19.
    • III. 20 -- 22.
    • III. 26 -- 29.
    • III. 31 -- 32.
  • here is the homework assignment due Monday, Dec. 1. and here are solutions to this assignment. There appeared to be some confusion on some of the homework problems in this set, mostly misunderstandings on what to prove and what tools are allowed. Please study the solutions.
  • here is the homework assignment due Monday, Dec. 8. For this set, you need to do reading of Book III of the "Elements" and some reading of my notes on parallel postulate, and here are solutions to this assignment.
  • Here are some practice problems for the final exam. Here are solutions to select problems from the practice problems for the final exam.
  • Here are solutions to select problems from the practice problems for Midterm I--- slightly expanded on Dec. 18.
  • Here is the proposed formula sheet for the final exam. It is expected that you can work out 2x2 or 3x3 rotation/reflection matrices about coordinate axes/coordinate planes; those for more general situations will be provided when needed. For problems involving Euclidean proofs, you can support each of your steps either by propositions that are taught in a typical high school geometry course, or by referring to the propositions in Book I or III of Eulicd's "Elements". I will provide a short list of the relevant propositions for such problems. Please refer to the practice problems for the final exam for a list of propositions that are more closely relevant for our course.
  • Here is the distribution of scores on the presentations: 7 7 7 7 7 7 7 7 7 7 7 7 7 7 6.75 6.75 6.75 6.75 6.5 6.5 6.5 6.25, and here is the distribution of scores on class participation: 3 3 3 3 3 2 2 2 2 2 2 2 2 2 1 1 1 1 1 0 0 0.
  • Additional office hours during the final exam period are: 10:00am--noon, Thurs., Dec. 18, and 11:00am--noon, Fri., Dec. 19, both in Hill 230.