## Linear Algebra for Applications

Last updated Mon 17 Dec 2018

SKIP PAST THE ANNOUNCEMENTS AND WELCOME LECTURE if you so desire. ._. (or to near the bottom)

#### ANNOUNCEMENTS

• Dec. 17: You should now be able to see your grades. The treatment of letter grades by eCampus is strange; it seems to use a scale of 1 to 5 instead of 0 to 4, so the statistics it reports are hard to interpret. Anyway, there are 10 As and 5 Bs. Curve? Of course! The grades are about one letter higher than the numerical totals. The latter is the sum of two tests, homework scaled to a max of 250, and a few points for participation in the discussions. I corrected two minor errors in the test key.
• Dec. 11: Exam key is now installed below (revised Dec. 15).
• Dec. 4: Another message about the final exam.
• Dec. 1: Here is the final exam announcement (e-mailed yesterday).
• Nov. 26: The MathCAD output is now visible.
• Nov. 23: Here is the final exam key that belongs with the two midterm tests I posted earlier. Please note:
• There is an accompanying MathCAD worksheet that I can't post until I get it scanned (after the holiday weekend).
• There was also a take-home part. That won't happen this year.
• The second midterm that year came later than the midterm this year. So, some topics that may be covered heavily on the final appear on the old Test B.
• Nov. 5: I have given up on drawing pictures in TeX for this year. Near the bottom (starting with Week 9) you will find scans of my old hand-drawn figures. With this, I think that the lecture notes are complete.
• Oct. 28: Tests are graded. Grades will appear in eCampus tomorrow. The solution key is posted below. I will investigate how best to return your graded papers, but be warned that the amount of explanation I wrote on them depends on how high your paper was in the stack when I graded the problem in question; you may find the key more informative.
• Oct. 11: Test procedures (one more increment): The test has been uploaded and will be accessible by you at midnight tonight. Start from the new menu item "Exams" in the column on the left of your eCampus screen (above "Course Materials"). Be prepared to click through an annoyingly large number of layers of links, variously labeled "Midterm" or "Begin" or "testa". The test has 6 questions, but eCampus thinks it has only 1, because they are all in one PDF file. When you are ready to submit, use the button "Browse my computer" to get to the file you need to upload.
• Oct. 10: Test procedures (cont.): I should have made clear that this test is intended to be "closed book"; that's impossible to enforce, but you are all mature and honorable people. The issue, of course, is fairness; I don't want any misunderstandings of what is allowed. You can bring in as much blank paper as you like, but no books, notes, programmed calculators, Matlab etc., Wikipedia and Google etc., phone calls, e-mail, and anything I've forgotten -- just like a classroom test. Also, please see my post in the "review?" discussion thread about test content: Sec. 18 yes, Secs. 24 and beyond, no.
• Oct. 4: Test procedures: The midterm test will be on-line during the period Oct. 12-14 (Friday-Sunday). My understanding is that the deadline is 11:59 p.m. (23:59) CST (not CDT) Sunday, but I urge you to finish well in advance to be safe. Once you log in to the test, you have 2 hours to submit your responses. I planned the test to take roughly an hour, but the extra time allows for technical matters like scanning (see below). (The final exam will aim for 2 hours and allow 3.) You will type and submit your responses just like homework. But what if you need to include something handwritten or drawn? If possible, scan it and submit it as part of the document. If that's not possible (for technical or time reasons), take a photo with a mobile phone and e-mail it to me (fulling@math.tamu.edu).
• Oct. 3:I just realized that we didn't have any homework on change of basis and how it affects the matrix of a linear function. It seems that that, like row reduction, is something B&W assume everybody already knows. Before the test you might want to review Secs. 4.4 and 4.5 in "Linearity". (Earlier, somewhere in the discussion forum, I also recommended Sec. 5.3, but that is less pertinent to the imminent test.) Stay tuned for the promised message about the test-taking procedures; it will come tomorrow or later today.
• Oct. 2: Here are the solution sheets for 1994 tests in late September and early November. Caveat emptor: some circumstances are different.
• Sep. 14: The materials for Weeks 4 and 5 have been revised from the rough draft you may have seen earlier.
• Our grader is not a TA; he is paid for a limited number of hours per week to grade homework. Please do not attempt to contact him outside of eCampus. If there is a grading problem that should not be shared with the whole class, e-mail me and I will forward the matter to the grader if appropriate.
• The best way to approach homework frustrations is to discuss the problems as a group, and you are already doing that very well in eCampus. The grader and I will read the discussions but respond in, let's say, a restrained fashion.
• Please try to keep the discussions organized: Posts that are not directly related to homework assignments should go into one of the subject-matter related forums or into a new forum requested via the "topic request" forum. (Personal chat should stay in "Let's get acquainted".)
• Aug. 21: I picked two weekends for the tests. Come back here at the end of the week for some actual math!
• Aug. 17: Rough draft of the handout is now available. I haven't filled in the dates of tests, because I need to harmonize them with my other course.
• Aug. 15: Please read the course handout (linked below) when it becomes available. Meanwhile, here is some basic information:
The textbook is (part of)
• R. M. Bowen and C.-C. Wang, Introduction to Vectors and Tensors, 2nd edition.
This is a Dover paperback in which the original two volumes are bound as one. We will be using "Part II", which must not be confused with "Volume 2"; Volume 2 consists of Part III! You can probably skip Part I, which covers things that every math student either knows already or will soon encounter elsewhere.

#### COURSE PROCEDURES AND RESOURCES

• Course handout
• This is a Distance class, so there is no classroom or meeting time. I'll answer phone calls during office hours (which you'll have to share with undergraduates coming in person).
• Instructor: S. A. Fulling
Email: fulling@math.tamu.edu
• In addition to this web page, we will use eCampus.
• Test keys

#### WEEKLY "LECTURES" AND ASSIGNMENTS

1. Aug 27-31: Welcome to "Linear Algebra for Applications". The purpose of the course is to provide background for courses in applied functional analysis, such as Math. 642, and applied differential geometry.

Crudely speaking, one can say that linear algebra is the study of finite-dimensional, flat things (vector spaces and the linear operators that act on them), while functional analysis studies infinite-dimensional, flat things and differential geometry studies finite-dimensional, curved things. They are the natural generalizations, of which linear algebra is the foundation.

However, it is not quite correct to say that linear algebra is just finite-dimensional; it deals with those things that are true of all vector spaces, finite or infinite in dimension. But those purely algebraic concepts don't get one very far in studying infinite-dimen-sional vector spaces. Topological concepts (convergence) need to be introduced. That is what makes functional analysis. One of the main goals of this course is to move beyond the finite-dimensional focus of elementary linear algebra courses into the setting of infinite dimensions.

As far as linear algebra, in the strict sense, is concerned, this course will be ``in principle, rigorous''. That is, all concepts will be precisely defined, and the proofs of theorems will be either

• (a) given in class, or
• (b) available in the textbook, or
• (c) so straightforward and uninstructive that there is no point in making them explicit.

On the other hand, in discussing infinite-dimensional spaces it will often be necessary to allude to convergence concepts. Necessarily there will be some vagueness and handwaving at these points. They will be included in the lectures for orientation and motivation, with the understanding that a proper treatment is left for later courses in functional analysis.

It is assumed that everyone here has had a first course in linear algebra. Our emphasis is on things that are usually not covered in the undergraduate courses. Nevertheless, the first few weeks will spent in reviewing elementary linear algebra -- partly because people tend to forget things, but also to take the opportunity to generalize and deepen your understanding of the basics. For example, I want to underline that most things learned for finite-dimensional real vector spaces are also true of infinite-dimensional or complex spaces.

We'll start with some homework (see below) to review the elementary calculations with matrices; no formal lectures on that should be necessary, but I (and your classmates, I hope) will be happy to answer questions on eCampus. At this point my old notes say, "More advanced computations (determinants, row reduction) will be reviewed later, when the need arises," but I can't see anywhere in the notes where that promise was fulfulled. Bowen & Wang seem to have never heard of row reduction (Gaussian elimination), which is essential for doing hand calculations with matrices efficiently. Let me know if you want me to go into that in more detail. Meanwhile you can look at Chapter 2 of my undergraduate textbook (see link above).

If you have not yet looked at our eCampus page, please do so now. Click on "Discussions" in the menu on the left, then on the Forum title, "Let's get acquainted". To add your own short bio to the forum, click on "Create Thread". (You have the power to create new threads, but not to create new forums.) To reply to somebody else's post, click on "reply" within that post rather than making a new thread. I'll make a new forum every week or two to hold substantive discussions about the math.

In both lecture notes and homework you'll frequently see references to "Milne". That means the book (good, but now out of print) by R. D. Milne, Applied Functional Analysis: An Introductory Treatment.

Lecture notes on basic definitions about vector spaces.

Homework 1 (due Wednesday, Sept. 5) Instructions for how to submit the papers will be forthcoming soon.

Lecture notes on bases. (These will take us into next week.)

2. Sep 3-7:

Lecture notes on direct sums.

Homework 2 (due Wednesday, Sept. 12)

3. Sep 10-14:

Lecture notes on inner products.

Lecture notes on orthogonality.

Homework 3 (due Wednesday, Sept. 19) Henceforth, references otherwise unidentified will always be to Bowen and Wang's book.

4. Sep 17-21:

Lecture notes on linear operators.

Lecture notes on kernels and ranges.

Lecture notes on isomorphisms and projections.

Homework 4 (due Wednesday, Sept. 26) The last two problems set up an example of Fredholm theory involving differential equations, to be exploited next week.

5. Sep 24-28:

Lecture notes on adjoints and Fredholm operators.

Homework 5 (due Wednesday, Oct. 3)

6. Oct 1-5: I'm trying to give you the maximum flexibility this week and next (around the test), providing the reading early and delaying the homework collection.

Lecture notes on domain technicalities. (Necessary for full disclosure; don't worry about it this semester.)

Lecture notes on hermitian and unitary operators.

Lecture notes on preliminaries to spectral theory. (These will take us into next week.)

7. Oct 8-12:

Prepare for a test next weekend (Oct 12-14).

Homework 6-7 (due Wednesday, Oct. 17)

8. Oct 15-19:

Lecture notes on the finite-dimensional spectral theorem.

Lecture notes on an alternative proof.

If you have time, start reading next week's notes on factor spaces.

Homework 8 (due Wednesday, Oct. 24)

9. Oct 22-26:

Lecture notes on factor spaces

Lecture notes on elementary de Rham cohomology as another example of factor spaces (A few of the diagrams in these notes are still not finished. Here they are in pen.)

Homework 9 (due Wednesday, Oct. 31)

10. Oct 29 - Nov 2:

Lecture notes on Jordan canonical form

Lecture notes on proof of the Jordan theorem (Look here for the diagrams that are missing handwritten annotations.)

Homework 10 (due Wednesday, Nov. 7) (Election Day special -- not in the sense that you can wait till Wednesday to vote, but that the assignment is so short you will have plenty of time to vote on Tuesday)

11. Nov 5-9:

Lecture notes on functions of an operator

Lecture notes on other applications of Jordan theory

Homework 11 (due Wednesday, Nov. 14)

12. Nov 12-16: Antisymmetric operators

Lecture notes on antisymmetric operators and related matter (A missing diagram is here.)

Homework 12 (due Wednesday, Nov. 28 [sic])

13. Nov 19-28: Dual spaces

Lecture notes on linear functionals and dual spaces

Homework 13 (due Wednesday, Dec. 5)

14. Nov 30 - Dec 5: Tensors

Lecture notes on bilinear forms

Lecture notes on general tensors (The missing diagrams are here.)

Go to home pages: Fulling ._._. Calclab ._._. Math Dept ._._. University

e-mail: fulling@math.tamu.edu