This is the main webpage for TMA4145, Linear Methods, for autumn 2010. On this page, you will find information about the course. More detailed information about the contents of the course can be found on the course wiki. There will also be a discussion forum for questions relating to the course.

The most basic object of study in mathematics is of a process. Processes take in input and spew out output. Almost everything that one wants to study scientifically can be modelled mathematically by a process. Given a process, there are three types of question that one wants to answer:

1. What I put in X, what did I get out?
2. Where I got out Y, where did I start?
3. How I put in X, I got out Y, how did I get it?

We model processes in mathematics by functions. Thus the study of processes leads one, in mathematics, to the study of functions. In the first two questions, the function is (presumably) known and the questions are about studying it. The last question in the list above is concerned with the question of finding a function. To do this, one must have some idea of the types of function that might fit, how these behave, and how to describe them. This involves studying not just one function but whole families of functions.

One of the simplest such cases is the family of continuous functions on the interval. One can think of these as modelling processes whereby:

1. The input is a parameter between 0 and 1.
2. The output is a (real or complex) number.
3. The output depends continuously on the input.

In this course, we shall develop the tools necessary to study such functions and, more importantly, the space of such functions.

Due to the nature of this course there is no single text which we follow. It is therefore not possible to assign a specific textbook for this course. The following is a list of recommended reference books for this course.

In order of use:

• Introductory Functional Analysis with Applications. Kreyszig, John Wiley & Sons. Selected material from chs 1 and 5 will be available as a handout from the department office (cost: 20Kr).
• Linear Algebra. Friedberg, Insel, and Spence, Pearson (ISBN: 0130084514).
• Linear Algebra by Jim Hefferon.
• A First Course in Linear Algebra by Rob Beezer.
• An introduction to Hilbert Space. Young, Cambridge University Press (ISBN 0-521-33717-8). Chs 1-4, 6

Some comments on the texts and their use:

• These books are recommended as reference books rather than text books. This means that they should be viewed as sources for supplementary material to help you understand the material rather than as the primary source.

• The second two linear algebra books are free (see the respective pages for exact details of their licenses), but the first contains more of the material from this course.

• Although the Hilbert space book is an excellent one, we only use a small amount of it. Most of the material on Hilbert spaces is also contained in the first Linear Algebra book. Therefore, if you expect to use Hilbert spaces beyond this course then I recommend that you obtain the book by Young. If you expect that you will not use Hilbert spaces further, then I recommend that you obtain the book by Friedberg et al..

Main Page

Technology permitting, the lectures will be delivered as PDF presentations. I shall endeavour to make the notes available the day before the lecture so that students can "follow along". Assuming that the technology works, there may be additional notes written during the lectures. These will be posted shortly after the lecture finishes. I shall often prepare a little more than I shall actually give, any extra will be taken up in the next lecture or deferred to the wiki.

The notes will be available in several different layouts. It is important to know which is which.

1. Beamer. This is what will actually appear on the screen during the lectures. You must never print this version. As each "transition" results in a new page, this can easily exceed 100 pages.

2. Trans. This is a "one frame per page" version of the above. If you intend to "follow along" with the lecture on your own computer then this is probably the best. However, I still strongly urge you not to print it.

3. Handout. This is a more condensed version in terms of space. By putting 4 slides on a page the total number of pages is significantly reduced. If you want to print something then print this version.

4. Annotated. It will probably take a little experimenting to find the best way to present the annotated version. As a first go, the PDF contains just the annotated pages. For most pages, this should be sufficient to locate it within the main presentation. For some it may be useful to have the previous page included as well. Let me know if this, or something else, would be useful.

Note: all are as PDFs.

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1. 23rd August. Introduction. beamer trans handout annotations

2. 24rd August. Functions. beamer trans handout annotations

3. 30th August. Calculating with Calculators. beamer trans handout annotations

4. 31st August. Producing Polynomials. beamer trans handout annotations

5. 6th September. Complete Convergence. beamer trans handout annotations

6. 7th September. Completely Fixed. beamer trans handout annotations

7. 13th September. From Here to There. beamer trans handout annotations

8. 14th September. Inheritance. beamer trans handout annotations

9. 20th September. An Angular Attack. beamer trans handout annotations

10. 21st September. That Which We Call a Vector. beamer trans handout annotations

11. 27th September. The Point of Angles. beamer trans handout annotations

12. 28th September. The Right Angle. beamer trans handout annotations

13. 4th October. The Luxury of Linearity. beamer trans handout annotations

14. 5th October. Dimension Leap. beamer trans handout annotations

15. 11th October. Rank Invariants. beamer trans handout annotations

16. 12th October. There and Back Again. beamer trans handout annotations

17. 18th October. How to Solve It. beamer trans handout annotations

18. 19th October. The Return of the Angle. beamer trans handout annotations

19. 25th October. Factors and Angles. beamer trans handout annotations

20. 26th October. Polynomials. beamer trans handout annotations

21. 1st November. Everything You Ever Wanted to Know about Hilbert Spaces. beamer trans handout annotations

22. 2nd November. Bases and Isomorphisms. beamer trans handout annotations

23. 8th November. Near and Far Away. beamer trans handout annotations

24. 9th November. Duality. beamer trans handout annotations

25. 15th November. Continuous and Linear: Best of Both Worlds. beamer trans handout (no significant annotations made)

    Note: This lecture will be shorter than usual; starting at the usual time but ending early.

26. 16th November. Spectral Theory. beamer trans handout annotations

    Recall: There will be no lecture on 22nd November. Instead, there will be a revision lecture in week 48 (most likely in the usual slot on 30th November).

27. 23rd November. Revision I. beamer trans handout annotations

28. 30th November. Revision II. Location: KJL1. annotations

Main Page

Links to official information about this course:

• Description Norsk English
• Final exam dates (scroll down a long way)
• Time plan.

Note: The times and places for the øving sessions will need altering.

TMA4145h2010

There will be no problem session (øving) in week 34. The first problem sessions will take place in week 35.

The times, locations, and session leaders for the problem sessions are:

• Wednesday 8:15-10:00, B3; Geir Bogfjellmo
• Thursday 15:15-17:00, S21, S22; Magnus Norling

Homeworks are due in on Fridays by 3pm in the boxes in the Nordre Lavblokk, 3rd floor which are marked as for this course. Please list the members of your group on the copy that you hand in.

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You are to do the homeworks in groups of about 4 students. The groups are now on the wiki. If you are not yet in a group, please sign up in accordance with the instructions there.

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• Homework 1 PDF due in Friday September 3rd. Solutions now on the wiki.
• Homework 2 PDF due in Friday September 10th. Solutions now on the wiki.
• Homework 3 PDF due in Friday September 17th. Solutions now on the wiki.
• Homework 4 PDF due in Friday September 24th. Solutions now on the wiki.
• Homework 5 PDF due in Friday October 1st. Solutions now on the wiki.
• Homework 6 PDF due in Friday October 8th. Solutions now on the wiki.
• Homework 7 PDF due in Friday October 15th. Solutions now on the wiki.
• Homework 8 PDF due in Friday October 22th. Solutions now on the wiki.
• Homework 9 PDF due in Friday October 29th. Solutions now on the wiki.
• Homework 10 PDF due in Friday November 5th (NB: the version on the wiki has some hints for question three). Solutions now on the wiki.
• Homework 11 PDF due in Friday November 12th. Solutions now on the wiki.
• Homework 12 PDF due in Friday November 19th. Solutions now on the wiki.

TMA4145h2010

• Course wiki
• Course Forum
• math.SE a question-and-answer site for mathematics of all levels

Final Exam:

• English
• Norwegian
• Solutions

The referance group for this course consists of:

• Jørgen Endal j.endal at hotmail dot com (MTFYMA)
• Stein-Olav Davidsen steinoda at stud dot ntnu dot no (MTFYMA)
• Sigurd Store storves at stud dot ntnu dot no (MTEL)