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Linear problems occur throughout mathematics and its applications. This course will examine the mathematical tools needed to study such problems. The simplicity and elegance of these tools is part of what makes linearity such a desirable property. Indeed, many problems that are not inherently linear are often made linear as a first step in their analysis.

Problems that require only a finite number of pieces of information to specify them can be reduced to the study of matrices and coordinate vectors. However other problems, such as the evolution of a sound wave, require so-called infinite dimensional analysis. In this course we shall study both.

The topics that we shall cover include:

• Approximation and metric spaces
• Linear transformations and their characteristics
• Infinite dimensional linear spaces

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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 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 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

Note that the two linear algebra books are free (see the respective pages for exact details of their licenses).

Although the Hilbert space book is an excellent one, we only use a small amount of it. Therefore I do not recommend that you buy it just for the sake of buying a book. It will not be necessary to buy it to take this course. In particular, other documents and resources will be made available when we are covering that material. On the other hand, if you are looking for a book on Hilbert space theory then it is an excellent resource to have.

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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. 18th August. Mastering Mathematics. beamer trans handout (no significant annotations made).

2. 20th August. Fun with Functions. beamer trans handout (no significant annotations made).

3. 25th August. Meaningful Measurements. beamer trans handout annotations

   Note: we did not get through the whole of this in the lecture. We got to the slide "Story so far".

4. 27th August. The Art of Approximation. beamer trans handout annotations

5. 1st September. Jogging Jacobi. beamer trans handout annotations

6. 3rd September. Notorious Neighbourhoods. beamer trans handout (no significant annotations made).

7. 8th September. Simply Sequential. beamer trans handout annotations

8. 10th September. Bumping into Banach. beamer trans handout annotations

9. 15th September. Case Studies. beamer trans handout annotations

   Note: We did not get through the whole of this lecture. We got to the slide "Examples".

10. 17th September. Continuous Case Studies. beamer trans handout annotations

    Note: We did not get through the whole of this lecture. We got to the slide "Examples".

11. 22nd September. Amazing Angular Abilities. beamer trans handout annotations

    Note: We did not get through the whole of this lecture. We got to the slide "Examples".

12. 24th September. Orthogonality. beamer trans handout annotations

13. 29th September. The Luxuary of Linearity. beamer trans handout annotations

14. 1st October. Marvellous Matrices. beamer trans handout annotations

15. 6th October. Dimension. beamer trans handout annotations

16. 8th October. Rank Invariants. beamer trans handout annotations

17. 13th October. Serious Simplicity. beamer trans handout (No annotations due to technical fault.)

18. 15th October. Fun with Factorisation. beamer trans handout annotations octave demonstration

19. 20th October. Terribly Triangular. beamer trans handout annotations

20. 22th October. The Return of the Angle. beamer trans handout annotations

21. 27th October. Furious Factorisations. beamer trans handout annotations

22. 29th October. The Final Factorisation. beamer trans handout annotations

23. 3rd November. Back to Bas(ic|e)s. beamer trans handout annotations

24. 5th November. A Series of Orthonormal Events. beamer trans handout annotations

25. 10th November. Close Encounters of the Linear Kind. beamer trans handout annotations

26. 12th November. Duality. beamer trans handout annotations

27. 17th November. Adroit Adjunctions. beamer trans handout annotations

    Note: due to the overlap with the Lars Onsager lecture, the second half of this lecture was cancelled. We were just about to start on the Examples.

28. 19th November. Adroit Adjunctions Again And Adieu. beamer trans handout annotations

TMA4145h2009

This course has a wiki. The intention is that the wiki forms a bridge between the lectures and the books. Both lectures and books by themselves have limitations: lectures are of fixed duration whilst books are not specific to this course. The wiki will serve as a midpoint between the two. One function will therefore be as a place for me or others to write supplementary notes.

By virtue of it being a wiki, though, it will also serve as a place where students can interact with the lecturer and assistants.

This is somewhat of an experiment and so any suggestions on how best to use this will be considered.

The wiki displays XHTML+MathML+SVG. This means that your browser might need tweaking in order to get the pages to display correctly. Detailed instructions are here.

For those interested, the wiki is instiki and uses markdown and itex to produce MathML.

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This course has a reference group. This is a way for those taking the course to give feedback while the course is in progress. The reference group will meet regularly with me to pass on (anonymously) any comments that they have received. This is the best way to pass on your comments on the course so that I can take them into account during the course. Both positive and negative comments are welcome.

The reference group are:

• Haakon Bakka bakka@stud.ntnu.no
• Simen Lønsethagen simenal@stud.ntnu.no
• Børge Solli Andreassen borgesol@stud.ntnu.no

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Links to official information about this course:

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

Note: The øving has been extended, rescheduled, and relocated. Beginning in week 35, it will take place on Wednesdays from 14:00 to 16:00 in the following rooms:

• B1
• EL4

TMA4145h2009

There will be a revision session in S4 on Monday 14th December from 2pm to 4pm. There is a page on the wiki for recording in advance any questions that you would like covered in that session.

More Messages

Here are some homework assignments from the previous two years, with solutions, together with some old exams, again with solutions. Remember that although the core material stays the same, the focus can vary from year to year. Therefore use these carefully. Remember that they weren't designed explicitly for this course given this year.

Assignments (2008)

• Assignment 1 PDF. Solution PDF.
• Assignment 2 PDF. Solution PDF.
• Assignment 3 PDF. Solution PDF.
• Assignment 4 PDF. Solution PDF.
• Assignment 5 PDF. Solution PDF.
• Assignment 6 PDF. Solution PDF.
• Assignment 7 PDF. Solution PDF.
• Assignment 8 PDF. Solution PDF.
• Assignment 9 PDF. Solution PDF.
• Assignment 10 PDF. Solution PDF.
• Assignment 11 PDF. Solution PDF.
• Assignment 12 PDF. Solution PDF.

Assignments (2007)

• Assignment 1 PDF. Solution PDF.
• Assignment 2 PDF. Solution PDF.
• Assignment 3 PDF. Solution PDF.
• Assignment 4 PDF. Solution PDF.
• Assignment 5 PDF. Solution PDF.
• Assignment 6 PDF. Solution PDF.
• Assignment 7 PDF. Solution PDF.
• Assignment 8 PDF. Solution PDF.
• Assignment 9 PDF. Solution PDF.
• Assignment 10 PDF. Solution PDF.
• Assignment 11 PDF. Solution PDF.
• Assignment 12 PDF. Solution PDF.

Midterms

• Midterm from 2004 PDF. Solution PDF.
• Midterm from 2005 PDF. Solution PDF.
• Midterm from 2006 PDF. Solution PDF.
• Midterm from 2007 PDF. Solution PDF.
• Midterm from 2008 PDF english. PDF bokmål. Solution PDF.

Finals

• Final from 2000 PDF. Solution PDF.
• Final from 2001 PDF. Solution PDF.
• Final from 2002 PDF. Solution PDF.
• Final from 2003 PDF. Solution PDF.
• Final from 2004 PDF. Solution PDF.
• Final from 2005 PDF. Solution PDF.
• Final from 2006 PDF. Solution PDF.
• Final from 2007 PDF english. PDF bokmål. Solution PDF.
• Final from 2008 PDF english. PDF bokmål. Solution PDF.
• Continuation exam from 2008/2009 PDF english. PDF bokmål. Solution PDF.
• Final from 2009 PDF english. PDF bokmål. Solution PDF.
• Continuation exam from 2009/2010 PDF english. PDF bokmål. Solution PDF.

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The definitive version of each homework is the MathML version on the wiki. However, to ensure that everyone can access it, there is a PDF there and another will be listed here.

• Homework 1 PDF

  The version on the wiki now has solutions as well.

• Homework 2 PDF

  The version on the wiki now has solutions as well.

• Homework 3 PDF

  The version on the wiki now has solutions as well.

• Homework 4 PDF

  The version on the wiki now has solutions as well.

• Homework 5 PDF

  The version on the wiki now has solutions as well.

• Homework 6 PDF

  The version on the wiki now has solutions as well.

• Homework 7 PDF

  The version on the wiki now has solutions as well.

• Homework 8 PDF

  The version on the wiki now has solutions as well.

• Homework 9 PDF

• Homework 10 PDF

  The version on the wiki now has solutions as well.

• Homework 11 PDF

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