Page last updated on Wednesday, December 17, 2003

Math 640 - 600/700 — Linear Algebra for Applications
Fall 2003
News

12/17/03. The solutions to the final exam have been posted. Check your neo account for the URL.

12/16/03. There's a new prime in town. See the story at http://www.foxnews.com/story/0,2933,105383,00.html. Also you may check out the primes page - yes there is one - for more details on big Mersenne primes see http://www.isthe.com/chongo/tech/math/prime/mersenne.html.

The new find has six million digits and was found just last month as a consequences of the GIMPS.
Is a number of this size really big? Well suppose we type on a page of paper 80 digits per line and 50 lines per page. Then, each page would contain 4000 digits. This gives a mighty tome of 1,500 pages to hold the new number.

... Thanks for this information go to Debbie Fannin and Mary Lou Shelton.

But is it really big. Like what number have we approximated to a lot of digits? The answer is pi. And pi is now known to 206,158,430,000 digits. Stringing them out, will take you to the moon and and part way back. (See, http://pw1.netcom.com/~hjsmith/Pi/Rec206.html)

12/10/03: The final will be posted sometime on Saturday. I'll give 72 hours to complete it. You should have your neo account active. I will send the link via that channel.

12/3/03: I've posted the solutions to homework set 10. Problem 5 was a bit tricky. You needed to work with two unitary matrices, not just one. I found the example, as written in some Russian text on matrices, and it did take a couple of minutes to see how to get those complex signs back in the mix after the D*D took them out.

Chapters 9 and 10 are now posted. I have posted a set of review questions for the final. See the link. This review sheet will be indicative of the level of the final exam questions; it is not the final itself. There is no answer key available. If you have particular questions about it, please email me directly.

12/1/03: Chapter 8 is now posted. The material of this chapter is, to say the least, among the most complicated and difficult material we have studied. Please give yourself a chance to understand it by not demanding of yourself full comprehension in just one or two readings. Study it carefully, trying to get the flow of the material before the detail. I strongly suggest you read the very long proof from two different sources. This theorem is, in a way, a culmination of all that we have been doing for the last half of the course.

There is no time for further homework assignments. I will roll all homework into the final exam.

11/21/03: Homework 10 is now posted. No due date has been set.

More information on the Tomball short course:
This course will focus on teaching algebra using visual technologies,
and learning how to make online assessments.

It will run 4 days, 8-5. In addition there will be a capstone project to be completed afterwards. You can earn up to 3 hrs (Math 685 or 696) credit for this. Limited scholarships may be available. Stay tuned for more information.

11/19/03: Chapter 7 is now posted

11/18/03: Solutions to problem set 9 have been posted.

11/17/03: Chapter 6 is now posted. A new video has been uploaded.

Coming this summer new course offering: Math 689 - Theory and practice of mathematical assessment. Summer session I.

Math 696 - Communication and Technology in Mathematics will be offered as a short course in Tomball, TX, May 17-20. Write for more details if you are interested.

11/13/03: Solutions to Problem set 8 are now posted.

11/9/03: NOTE: on Problem 3 of the current set, do not do part (iii).
I have finally managed to get the complete chapter 5 posted with the images.

...

11/6/03: The final exam is scheduled for December 15, Monday 3:30-5:30 p.m. Distance students will submit their final by that time.

11/5/03: The next homework assignment is posted. As I've converted it to pdf using another method, please let me know right away if you notice font problems.

11/4/03: The solutions to the midterm exam have been posted. (in class version --- distance version)

10/29/03: Your assignment for next Wednesday, 11/5, is to read the paper by MIT mathematician Gilbert Strang titled The Discrete Cosine Transform. It is located in the SIAM Review, 41, 1999, pp. 135-147, or online at http://epubs.siam.org/sam-bin/getfile/SIREV/articles/33674.pdf. The discrete cosine transform (DCT) is one of the many methods used to compress images. If you have heard of JPEG or JPG encoded images, you will note that the compression in them is partly based on one form of the discrete cosine transform. You should find that you are more than prepared to read this level material. Note the natural way the colums of the DCT matrix are shown to be orthogonal. Clever! BTW, Strang is the past president of SIAM, the Society for Industrial and Applied Mathematics.

Actual assignments on the DCT will be forthcoming.

10/28/03: The solutions for Problem set 7 have been posted.

10/27/03 This week: Section 600 - midterm exam is Wednesday in class.
Section 700 - link to exam will be sent on Thursday. Exam must be returned by Sunday at 6pm.
Exam coverage: Through sectoin 4.1.
Notes for Chapter 4 and 5 have been posted.

10/22/03 Clarification: the exam for distance students begins next Thursday, Oct 30.

10/22/03 The solutions to problem set 6 are now posted.

You need to write a term paper on some application of linear algebra. It may be computational linear algebra, or other area. Almost every mathematically technical subject uses matrix theory. So, you should find no trouble finding an application. Noentheless, here are some ideas.

10/21/03 Practice problems for the midterm exam. Note, there is no answer key for this one. The rule is that any of the problems or variations of may be asked on the exam, but there is no guarantee that only these problems will be used.

For distance folks, I will release your exam on Thursday. The link will be mailed to your neo account. It will be due by Sunday evening.

More videos have been posted. [Look below for information on Mt. Everest.]

Linear Algebra Application - Image compression. Using a technique called the singular value decomposition, we can compress an image to a small fraction of a size. Below we show several pictures of the famous "Lenna" image at various levels of compression. The level is regulated by how many eigenvalues to retain. We'll talk more on this later after finishing Chapters 4 and 5.

Lenna: Original image 147 x 137 pixels.

Original color image	Original BW version	Compression using 95/137 eigenvalues (96%)
Compression using 45/137 eigenvalues (21%)	Compression using 15/137 eigenvalues (2%)	Compression using 11/137 eigenvalues (1.3%)
The percentages give the relative amount of data required to render the displayed image.

Quantizers. In a second application of data compression we use a uniform quantizer, which means we group the data into compartments. For example suppose we have a range of decimal data between zero and ten. Suppose we use ten compartments. Then, for example, if a particular datum is between 5 and 6 we reassign it the value 5.5. In this way we replace possibly a large continuum of values by just 10 values. Then the data can be compressed for transmission in a number of ways. This quantization is called lossy because information is permanently lost.

Original BW version	using 51/250 ranges. Compression 5:1.	using 26/250 ranges. Compression ~10:1.
using 17/250 ranges. Compression ~15:1.	using 11/250 ranges. Compression ~23:1. This image uses eleven shades of gray.	using 6/250 ranges. Compression ~41:1. This image uses only six shades of gray.
The percentages give the relative amount of data required to render the displayed image.

There are several types of quantizations including the variance and variable rate quantizers.

10/20/03. Here's a question. How high is Mount Everest? We all know this. But... How and when was the elevation first computed, and what was it like to be on the survey project team? Read the story of who contributed what at the following link. Also, is the elevation the same today as it was back in 1856 at the time of the first announcement? http://news.bbc.co.uk/2/hi/south_asia/3193576.stm
It's not linear algebra but it sure is interesting applied mathematics.

10/16/03. Problem set 7 (aka homework 6 continued) has been posted. Videos for Section 3.5 are posted, as well.

10/8/03. Problem set 6 is now posted. Note the due date if Oct 17. Solutions for problem sets 4 and 5 are also posted.

The midterm exam is scheduled for Oct 29. This will be an in-class exam for local students. Distance students will get a time limit three days and have a different exam.

10/8/03. Problem set 5: Hint for problem 8. You will notice that in this problem two propositions must be established, one for even n and the other for odd n. So, if you plan to use induction, it seems natural to base the induction on the separate propositions. Thus, there will be two induction proofs.

Hint for problem 5. Don't compute things. Use the results from section 2.4.1. With that result and the single computation required, this problem becomes trivial.

The sandbox origin of Problem 8: Basically, I was playing around with Maple constructing the types of matrices in the problem computing various quantities as the size changed. Then I noticed the remarkable property of the determinant. Of course, it was no proof. But it did establish a firm "belief" in the validity. Then I proved it.

There is something strangely psychological at work that I have never fully understood. To prove a difficult proposition is true, you must first truly believe it to be true. Conversely, to proof a difficult proposition is false, you must first truly believe it to be false.

10/6/03. New videos for Chapter 3, Sections 1 and 2 have been posted.

10/5/03. Here are some hints for the Homework set 5.

Also. I did indeed have a hard disk crash. What I lost is my entire email archive, a research paper I'm working on and other files. I also lost a really problem set for this course which was almost complete. How does this impact you? Well, if you have been faxing your homework in, I want to mail it back to you. Therefore, please resend your address.

10/1/03. A number of you have having trouble making these changes of bases. Do not fret. For some reason this topic usually causes a lot of trouble. Here's some extra help on changing bases.

9/30/03. Don't forget. NOVA will air a program on the Archimedes Palimpsest. THAT's TONIGHT!!!

Midterm exam: Oct 29.

Homework 5 is posted.

9/30/03. Disaster has struck. I've had a hard disk crash - kinda. Actually, the drive is in the process of failing. Instead of pushing the last little bit out of it, I took it in for repair hoping that the CIS gurus could save what was on it by making an image. The main damage amounts to most of my email archives and other stuff. We are currently checking on the status of homework. I may ask some of you to resubmit past work.

9/28/03. At the following link is a status of topics in this course and which we have covered. You will find quite a bit has been accomplished so far. We have a long way to go with most of the interesting and sometimes surprising results yet to come.

9/27/03. New videos have been posted. Check out the video page. The hints are now also in video viewing form.

9/26/03. The homework due date for homework 4 is changed to Thursday, October 2.

9/26/03. Here are some hints for the homework questions.

1. How can I find a matrix representing a differential operator, T, on a function space? Link. - Link (in pdf)
2. How can I show that the repeated application of various rotations is an invertible operator. That is, how can I show the matrix representation is invertible. Link. - Link (in pdf).

9/24/03. Please to not send emails to the WebCT mail address. I will not likely read them in any reasonable time. If you need to send an email, sent it to dallen@math.tamu.edu.

9/24/03. Solutions to homework set 3 are posted. Note that these solutions are NOT the only way to solve the problems. Your solutions will be graded as correct or not, and not whether you can duplicate the solutions I prepared.

9/19/03. WebCT maintenance window on Sunday, September 21, 8:00a.m. - 10:00a.m.

9/16/03. I have posted an Appendix on solving linear system. Though I am sure you remember how to do it from your earlier courses, I added additional material. It is Appendix A of Chapter 2, now included with Chapter 2 and also as a separate appendix.

9/15/03. I've posted a couple more videos. They take you current with the course. Let me know by email if you want videos of the homework solutions.

On the homework 1 solutions. For the problem with the polynomials (#3), many of you correctly were able to show the set of vectors (polynomials) were linearly independent. But you didn't conclude that the set must be a basis, rather just stated it.

Norms: Several of you have had some problems with the homework problem with polynomials and whether |p(2)| is a norm. Here are a couple of points to clarify when working on a problem. (1) What the vectors actually are. [In this case they are functions, quadratics in particular.] (2) A norm is first function defined on the vectors. [In this case, the function is the absolute value of the evaluation of the quadratic at the value 2.] (3) A norm satisfies three additional properties. [In this case, the function is NOT a norm if there is a single vector for which it fails one of the properties. And all you need is one such vector.]

9/11/03. In Chapter 2 I state on about page 47 that the RREF is not unique. In fact, this statement is not true. Iin a slightly revised version of Chapter 2, this misstatement has been corrected. (The statement is a hold-over from an earlier version where I used a different RREF which is not unique. ) Sorry about that. I've given an interesting proof using a double induction. A double induction is a induction within an induction. In this case it is on the number of columns and inside the number of rows of the matrix.

Have questions? Post them to WebCT. The turn around is usually quick.

9/8/03. I know some of you have taken Math 629 or will be taking it. So, here is a good history of math. On Sept 30, NOVA will air a program on the Archimedes Palimpsest.
It's a good mathematics detective story.
For more information see,
http://www.pbs.org/wgbh/nova/archimedes/palimpsest.html
http://www.thewalters.org/archimedes/frame.html
http://www.cis.rit.edu/people/faculty/easton/k-12/exercise/
http://www.wikipedia.org/wiki/Archimedes_Palimpsest

Homework set 1 is due today.

9/7/03: Homework set 2 is posted.

Here is an interesting question related to problem 5 of homework set 1. We have two basis of a 2k vector space. Can these two bases be chosen so that taking k aribtrary vectors from each and combining them there results a basis of the entire 2k dimensional space?

9/4/03: I don't know about you, but... I have just discovered that if I put the URL https://webct.tamu.edu/SCRIPT/math_640/scripts/serve_home into my browser, I get into WebCT without needing to go through the login process. Give it a try. However, if you have closed your browser and then reopened it, you may have to re-login.

Section 1.4 was in my view not presented clearly in a theorem-proof fashion. This extension theorem states that we can always "build-up" a linearly independent set to a full basis if we have at hand a finite spanning set.
I changed the presentation slightly to make it appear more orderly. It is online and integrated into Chapter 1.

9/3/03: To view the movies properly, you may need the Tech Smith code to properly view the movies. For PC's it is a free (and really small) download. Click here or here. To install: Double click on the program tscc.exe and it self-installs. If you are using a Mac and are having trouble viewing the videos please let me know.

9/2/03: For Spring '04 preregistration dates consult the university calendar at http://www.tamu.edu/admissions/records/academic_calendar.html.

The second set of videos has been posted. This includes the conclusion of Chapter 1 Section 5 and Chapter 1 Section 6. Also Chapter 1 has been reposted (again) with a few typos corrected. Please report to me any font problems. Thanks.

9/1/03: The first of the video lectures has been posted. You can obtain the link from the main page or click here. These video are made with video capture technology amount to a verbal walk through of the material begin covered in this course. You will need a media player to view them. The link to the Microsoft Video Media Player is on the video page. Let me know if you are having trouble. Thanks.

8/31/03: Chapter 1 notes have been reposted. Homework set 1 is not posted.

8/30/03: First steps: Open a NEO account. Go to http://neo.tamu.edu/. Having opened a neo account, you will receive broadcast messages I will be sending from time to time. We will be using WebCT for a portion of this course. Namely, we will have a discussion page for mathematical questions.
Accessing WebCT: (a) Connect to http://webct.tamu.edu, (b) Login using you neo account login ID and password. (c) Select Communication Tools, (d) Select Discussion, and finally (e) Select Main. This will get you access to the discussion board. You may look at all messages or just the unread message by choosing the appropriate button. If you enter a question give it an appropriate subject. Feel free to respond to posted questions. Just experiment with all the buttons. It's not difficult, but may seem daunting at first.

• Post your mathematical questions there. This include questions about problems, theory, homework, examples, and exams.
• I will post solutions to homework on the main site on the course homepage.

NOTE: If you have a (mathematical/course relevant) question it is likely that someone else does as well. So, don't be reluctant to post questions that pertain to the course.

Personal questions can be sent directly to my email address dallen@math.tamu.edu

Chapters 1, 2, and 3 are now posted. To read and print these files, you will need the Adobe Acrobat Reader.

**If you notice any font anomalies, let me know immediately. Thanks.