Math 2406, Abstract Vector Spaces, Fall 2012

Click here for a copy
of the syllabus.

And click here for a short intro to
symbolic logic.
Homework 1 (due Wednesday, Sept. 5)

Click here for that first homework.

For a note on cadinality, surjections/injections/bijections,
contable/uncountable, click here .
Homework 2 (due Wednesday, Sept. 19)

Page 130131, #2, 12, 23. Some notation: by "multiplication
table" he means that when you form a matrix of functions f along the
top row and functions g along the left column; and then the entries look
like f composed with g. And when he writes ST he means S composed with T.

Prove that if f : S > T is an injection, where S and T both
have the same number of elements, then f is also a bijection. Then
prove that the same holds true if f is a surjection.

Page 13, #1b, 2, 3.

Page 3031, #5, #12.

An announcement from ADAPTS: Note Taker Announcement:
A student note taker is needed in this course to take notes for a student
with a disability. The note taker will be paid a stipend for this
assignment. Skills needed are the ability to take accurate, legible, and
organized notes and a commitment to attend every lecture. If interested,
please contact Karishma Patel or Tina Allen via email at
notetaker@vpss.gatech.edu as soon as possible. Be sure to indicate the
Professor's name, time, day and course number in the subject line of the
announcement. Thank you!


Click here for a study sheet
for the exam.

I plan to hold a study session TOMORROW, THURSDAY Sept. 20 from
6:00pm until whenever in our classroom. See you there!
Homework 3 (due Wednesday, October 10)

Page 3637, #3,14,25.

Page 41, #1, 11, 14.

Page 4748, #10, 20.

Page 50, #6.

Page 57, #12.

Page 6263, #2, 3, 14.

Page 6768, #7, 8.

Page 76, #6, 20.

I recommend the following book if you want some practice problems
to look at: The Linear Algebra Problem Solver

When we start to work on matrices, the following online lectures
of Gilbert Strang might be helpful, as he is a brilliant lecturer: Click
HERE .
Homework 4 (due Wednesday, Oct. 31)

Page 9596, #4, 9, 11, 21, 25.

Page 102103, #5, 9, 21, 24. (pay special attention to 24)

Page 109110, #1, 3, 6, 15.

Page 118, #1, 5, 7.

Click HERE for a video on jpeg compression that I told you about in last
class.
Homework 5 (due with the exam, Nov. 12)

*** ***

Page 123, #1, 3, 5, 6, 7, 8, 10, 12, 13, 26, 29.

Page 139, #2, 5, 6, 11, 17.

Page 147, #14.

Page 158, #10.

Page 159, #6.

Page 174175, #1, 2.

Click HERE for a copy of the study sheet for
midterm 2

*** ***

Ok, so problem #2 on the exam was unfair, I think. When I
made the exam out beforehand it seemed like such an easy problem; but
it was much harder than I realized. So, I will give everyone 20 points
for that problem, regardless of what you wrote down. Actually, it turns
out to have an easy solution, which you can view HERE ; but it's unfair to expect people to come up with that on the exam; and I
apologize for that.

Here is an even more clever solution to problem 2: the condition
in the problem is equivalent to f(x+1)  f(x) = f(x+2)  f(x+1) for
all x; this implies that for any positive integer n we have
f(n)  f(0) = f(1)f(0) + f(2)f(1) + ... + f(n)  f(n1) = n(f(1)  f(0)).
So, f can grow at most linearly with n, which shows f can't have
quadratic (or higher) growth; and linear functions satisfy
f(x)  2 f(x+1) + f(x+2)=0,
which shows that the f's we're looking for are linear.

Problem 1 is the only other challenging problem on the exam
(at least computationally); for that one, I'll give massive partial credit.
If it looks like you have the right idea, I will give you almost full
credit; if you leave a few things unsimplified, I will give you a full
20 points. Remember, also, that I will curve the exam if the median score
is below 75. For a hurridlytyped note on problem 1, go HERE .

I have decided to implement a "maximal grading policy", whereby if
HW, MT1, MT2, and F are your homework, midterm 1, midterm 2, and final exam scores, then
your final grade will be max(F, (30%)HW + (20%)MT1 + (20%)MT2 + (30%)F).
Also, on your final I will present you with 15 problems and you will be allowed
to choose any 10 of them to complete  so, you will have more time per problem
than during the midterms; AND, you will have a choice on which problems to complete.
Homework 6 (due during the final exam)

Page 184185, #1, 3.

Page 191192, #1, 2, 3, 5, 6, 7, 8, 9, 10.

Page 202203, #5, 6, 7, 10.

Page 207208, #1, 4.

Page 214215, #1.

Page 219220, #2.

The Putnam exam is coming up, for those who are into math
competitions. I don't know how many spots are left for people wishing
to sign up for it; there may be some available. For those who don't know
what it is: it is a yearly intercollegiate math competition consisting of two
sets of about 10 problems each, where you are given something like 6 hours
to complete the problems. A large percent of people taking the exam
usually can't answer a single problem, despite how little math you need
to solve them  it's more a question of what you can do rather
than what you know (though extra mathmatical knowledge can't hurt you).
It used to be the case (and maybe still is) that if you place in the
top 10 you get an automatic trip to Harvard for math grad school; and those placing
in the top 100 or so will get a letter from the National Security Agency
asking them to come work for them (I'm being serious). There is a whole
culture built up around the Putnam exam  e.g. lots of stories involving
famous people like Richard Feynman, Norbert Wiener, and John Nash.

Click HERE for a page about the Google Pagerank algorithm (named after
"Larry Page", not "Web Pages").

There will be a study session this Thursday (Dec. 6) at 6:00pm in the
classroom.

Click HERE for a nice
writeup of the spectral theorem that we covered in class.

Click HERE for a study sheet for the final exam.



Here is an additional note.