12/12/11: The extra office hours will be tomorrow, Tuesday, December
13, 3:30-5:10 p.m. As with all office hours this week, if there are
too many customers, I will move the event to a larger room and leave a
note on the door of my office.
12/12/11: The extra office hours will be tomorrow, Tuesday, December
13, 3:30-5:10 p.m. As with all office hours this week, if there are
too many customers, I will move the event to a larger room and leave a
note on the door of my office.
11/29/11: The final exam is coming on Wednesday, December
14. Coverage: Sections 9.3-8, Chapter 10. How to get ready: the
more problems you solve before the text, the better you will be
prepared. Solve
the sample
exam, which will be posted at least a week before the test, and
problems from the past homeworks again (if you feel you do not
remember how to do them), as well as similar problems from the
text. Also, look for similar problems from College Geometry textbooks,
such as College Geometry by Howard Eves, available on reserve
in the Math Library, look for Professor Rogness's Math 5335 old exams
on the web. If you know you need only 3 minutes to get an idea on how
to solve at least 50% of random problems you find, then you may be
sure you will do well our test.
11/21/11: Some statistics for Exam II (out of 80 points total): mean =
50.5, median = 50, maximum = 80. It is amazingly close to the results
of Exam I, see below.
11/16/11: I have to cancel my office hours for Tuesday and Wednesday,
November 22-23, which is the Thanksgiving week. If you need to see me,
do not hesitate to make an appointment.
11/15/11: I will be holding extra office hours after 3:30 p.m. today,
Tuesday, November 15. Everybody is welcome to come.
11/14/11: Plans for Black Wednesday, November 23: this
pre-Thanksgiving class is canceled and will be made up through extra
office hours before the Final Exam. Happy Thanksgiving! No homework
will be due on Black Wednesday. Homework 9 will be due on Wednesday,
November 30.
11/09/11: As I have announced in class, I am removing Section 9.3 from
Exam II coverage, see below. You may also skip the 9-team geometry
from Section 9.2.
11/08/11: A hint for Problem 5.3.22: Consider two cases: (1) when the
triangle is acute or right and (2) when the triangle is obtuse. Argue
each way: => and <= separately.
11/01/11: The second midterm is coming on Wednesday, November
16. Coverage: Sections 7.7-8, Chapter 4 through Section 4.8,
Chapters 5, 6, 8, and 9 through Section 9.2, skipping the 9-team
geometry. How to get ready: the more problems you solve before the
text, the better you will be prepared. It is a good idea to complete
Homework 9 and be done with it, even though it is due only the week
after the test. Also solve
the sample
exam, which will be posted at least a week before the test, and
problems from the past homeworks again (if you feel you do not
remember how to do them), as well as similar problems from the text
(for example, if problem 11 was on the homework, problems 10 or 12
will likely be similar). Also, look for similar problems from College
Geometry textbooks, such as College Geometry by Howard Eves,
available on reserve in the Math Library, look for Professor Rogness's
Math 5335 old exams on the web. If you know you need only 3 minutes to
get an idea on how to solve at least 50% of random problems you find,
then you may be sure you will do well our test.
10/31/11: I have to move my office hours for Tuesday, November 1, from
11:00-12:30 to 10:30-12:00. Sorry about a short notice.
10/20/11: A hint for Problem 7.9.32: Recall that each isometry either
preserves or reverses orientation. This is material of Chapter 7
before Section 7.1. This should reduce the possibility for the
composition in the problem to only two isometries from the list. Then
try to come up with concrete examples that realize both
possibilities.
10/17/11: Some statistics for Exam I (out of 80 points total): mean =
50.5, median = 50, maximum = 80.
10/15/11: I have to move my office hours for Tuesday, October 18, from
11:00-12:30 to 3:30-5:00. If that does not work for you, either see me
at my office hours on Wednesday, 10:30-12:00 or make an appointment
for a different time.
10/12/11: Homework 5, due next Wednesday, October 19, has been
posted.
10/11/11: When we were going over Problem 5 from the Sample Test I
today during my office hours, we discovered I had made an error, when
adding the last translation in the solution to that problem I
presented in class on Monday. I have included the correct answer to an
updated version
of Sample
Test I. Sorry about the error in class -- a few points off my exam
score!
10/10/11: Some of solutions of Problem 7.9.7 from Homework 4 got
graded incorrectly, as the grader thought f composed with g
meant gf. I apologize about that. If you want me to regrade
that problem for you, please bring your graded homework to me, either
in class or during my office hours.
10/03/11: Three problems on Homework 4 have something to do with
reflections. Even though we have not covered those in class, I have
decided to keep these problems on the homework due on Wednesday. One
advantage for you is that this way you will get a chance to practice
with reflections and get those problems graded and returned to you
before the Midterm Exam. And all you need to know to solve these
problems is the formula U(X) = X - 2(<A,X> - c)A for a
reflection U in a line given by a special normal form
<A,X> - c = 0. This formula, written in a matrix form, i.e.,
when you pass to coordinates, will be a matrix formula for the
reflection.
10/02/11: The first midterm is coming on Wednesday, October
12. Coverage: Chapters 1, 2, skipping Sections 2.1 - 2.2, Chapter
3, Sections 7.1-5. How to get ready: the more problems you solve
before the text, the better you will be prepared. Solve
the sample
exam, which will be posted soon, and problems from the homeworks
again (if you feel you do not remember how to do them), as well as
similar problems from the text (for example, if problem 11 was on the
homework, problems 10 or 12 will likely be similar). Also, look for
similar exams, for similar Geometry classes at US schools on the
web. If you know you need only 3 minutes to get an idea on how to
solve at least 50% of random exam problems you find for similar
classes on the internet, then you may be sure you will do well our
test.
9/28/11: There is a shorter solution of Problem 3.8.26, which is
harder to arrive at. It is motivated by change of coordinates, of
which we know only a general idea from short, descriptive sections,
such as Section 3.7, of the text. Otherwise, this solution will look
like it is based more on a trick, rather than on common sense logic,
whereas the solution I presented in class was based on common sense,
even though it was quite long. Anyway, if you would like to see that
shorter solution,
click here.
9/27/11: Continuing the hint below for Problem 3.8.26. By finding an
isometry here we mean finding a matrix formula for this isometry U,
i.e., U(X) = M[X] + [P]. To figure out what M and [P] are, you need to
plot the given points and see what your isometry (a motion of the
plane) should be geometrically. Make sure to choose a simpler plane
motion out of all possibilities. Otherwise, your computations will be
harder to make. If you clearly know what U does geometrically, you
should be able to compute U(X) for any specific point, such as U(0),
U(-23,176). However, what you need is to find M and P. Use what you
know about isometries. One thing you know is P = U(0). So, see where
the origin maps to by your isometry U. It should be a very specific
point, such as (6,-2). (I can tell you that it will actually be
different.) This will be your P. How to figure out what M is? Well,
another property of isometries says that T(X) = U(X) - U(0) must be a
linear transformation. All we need to do is find the matrix of T, so
that T(X) = M [X]. There is a general recipe from linear algebra on
how to find matrices of linear transformations: the first column of M
must be T(1,0) and the second column of M must be T(0,1). (I reminded
that to you in class yesterday; this is also the formula just above
Equation (3.11) in the text.) Thus, all that remains to be done is to
find the images of (1,0) and (1,0) under T(X) = U(X) - P, where P has
already been computed, and you know where U maps points.
9/26/11: Sorry that I had to go over that example in class quickly, as
I was already running over time. Let me clarify the idea of writing
down a formula for an isometry when you know what kind of linear
transformation and translation (say, by a vector P) that isometry
should be made of. (You may also view this as a hint to homework
Problem 3.8.26.) When you figure out what that linear transformation
should do geometrically, it is easy to write down its matrix. Take the
standard unit vectors (1,0) and (0,1), see where they have to map to,
say, to vectors (a,b) and (c,d). (Make sure that these are orthogonal
unit vectors, so that the linear transformation is orthogonal.) Then
the matrix will be
| M | = |
|
, |
9/20/11: Hint to Homework 2 Problem 2.5.12: you can use either the
result of Proposition 11 or the proof of Proposition 11; if you use
the result theoreof, make sure to check that the assumption involving
l in the problem gives you the assumption involving the
interior in the proposition.
9/20/11: Hint to Homework 2 Problem 2.5.7: do not forget that (2,-1)
should be the vertex of the angle.
9/15/11: Hint to Homework 2 Problem 1.7.42: use
Cauchy-Schwarz. Regarding the Extra Problem on Homework 2: remember
that all we know in this logical development of geometry is what
arccos is.
9/14/11: If you are interested in the North Central and Putnam
mathematical competitions this semester, you should register for them
by the beginning of October. Interested students can contact Professor
Lawson at
tlawson at math.umn.edu to
register or be put on a mailing list for information.
9/12/11: Thanks to a couple of you for pointing out the following
misprint in the text. In Example 5 in Section 1.2, a minus sign is
missing in the answer. The correct answer should be (9/28,0) +
t(-12,-28).
9/8/11: If you are puzzled about what "special" means in reference to
Problem 1.7.21, search the subject index at the end of the text.
9/7/11: I have arranged that
some other
useful books be put on reserve in the Mathematics Library (310
Vincent Hall). These are not required texts, but rather recommended
readings, should you need extra reading material.
9/7/11: I have also added Section 1.4 to this week's reading
assignment. It is a good way to brush up on your Linear Algebra
skills.
9/7/11: I have updated coverage in the syllabus at 11:02
a.m. today. If you uploaded a copy before then, please correct that as
follows: Chapters 1, 2, skipping Sections 2.1 - 2.2, Chapters 3, 7,
skipping Section 7.6, Chapters 4 - 6, 8 - 11.
9/7/11: I recommend the following way to study. Attend each class,
take notes, participate in the discussion actively. After each class
review your notes and study the corresponding part of the text. You
can find out which part of the text at
the
Class Outlines page. Then do relevant
homework problems.
9/7/11: If you need to register for this class, please, send a message
to Ms. Lawson at ugrad@math.umn.edu and ask for a permission
number. Then go to OneStop and register.
9/7/11:
Homework will be collected before each class, starting next
Wednesday, September 14, and a small selection of problems will be
graded. ("Small," because the grader will allocate only 4 hours per
week for our class.)