The webpage for this course has RSS2.0 and Atom feeds. Click on one of the icons in the top left corner to subscribe (make sure that you are on a page relating to this course when you do so, otherwise you may get the feed for the entire site by mistake). The latest versions of the main browsers support feeds (sometimes called Live Bookmarks) or you can find a good free feed reader for your computer. It is probably best to get one that detects when an entry has been updated (rather than just one that looks for new entries).

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

Main Page

Lecturer: Andrew Stacey

Teaching Assistants: Steffen Junge and Eivind Fonn, eivindfo@stud.math.ntnu.no.

In order of use:

• Introductory Functional Analysis with Applications. Kreyszig, John Wiley & Sons. Selected material from chs 1 and 5; this is available as a handout from the department office (cost: 20Kr).
• Linear Algebra and its Applications, 4th edition. Strang, International Thomson Publishing (ISBN 0-534-42200-4). Chs 1 to 6 (omitting: 1.7, 2.5, 3.5, 5.3, 6.4, 6.5).
• An introduction to Hilbert Space. Young, Cambridge University Press (ISBN 0-521-33717-8). Chs 1-4, 6

Online resources (linear algebra):

• Wikipedia

  The following are all from the english version of wikipedia. I don't know what has been translated to the bokmål version.

  Useful articles:

  • Vector space

    This article contains most of the basic definitions of vector spaces, including some extensions such as normed vector spaces. Most of this article is relevant for this course, exceptions being "6.3 Tensor Product", "7.3 Algebras over fields", "9 Generalizations" (strictly speaking we have not and shall not discuss "quotient spaces" (from 6.1) either).

  • Linear map

    The basics of this article are contained in the "Vector space" article above, but this is more focussed. In particular, it contains a list of properties of linear maps such as "injective" and "surjective".

  • Basis and Dimension

    Of these two, the article "Basis" is the more detailed. The property that we have used most often (characterising finite dimensionality in terms of isomorphisms to Euclidean spaces) is mentioned in the "Facts" section of the "Dimension" article.

  • Linear independence and Spanning set

    These are not so extensive, but are useful expansions on the properties of bases.

  • Matrices and Transformation matrices

    Both are relevant but the second is more focussed on what is in this course.

  • Gauss Elimination

    The main computational tool of this part of the course.

  • LU Decomposition

    The algorithms (beyond Gauss Elimination) are not relevant.

  • Jordan normal form

    This deals with the complex case rather than the real one. The complex analysis is simpler due to the fact that there are no quadratic factors to consider.

  • Minimum polynomial

    This is not a very extensive article, I would recommend Axler's Down with Determinants instead.

  • Column space

    The part of this that is most relevant is the one involving finding bases for the column space.

  • Kernel and Kernel

    These are actually two articles, one focussing on linear operators and the other on matrices. The matrix one contains the details on finding bases.

• Down With Determinants

  This article is a great resource for the minimum polynomial and other concepts surrounding that, including the characteristic polynomial, generalised eigenvectors, and the Jordan canonical form.

  This article has since been expanded into a linear algebra text book which our library has available as an e-book.

• Planet Math

  This has many useful articles relating to linear algebra, but their quality is not as high as that on Wikipedia. The articles tend to be shorter and more general and thus less suited to this course. On the other hand, it tends to contain more proofs than Wikipedia.

• Factorisation Handount

  This is a handout I have written covering all the factorisations that we encounter in this course.

Main Page

The assessment depends on three factors:

• Assignments.

  There will be twelve of these in total. Eight must be passed for the candidate to take the final exam.

  To check how many of your assignments have qualified, enter your name in the following form.

  Name:

  For the assignments you will be divided into groups, each group to submit one copy of the solutions whose mark will count for all members of the group. The assignments are due in on Fridays in the marked boxes in Nordre Lavblokk, 3rd floor by 15.00. Make sure you put it in the correct box. The assignments must be clearly marked with the names of every person in the group.

• Midterm.

  The midterm will take place in week 42 on the 14th October in the usual lecture slot: 14:15 - 16:00.

  Locations:

  • Surname: A-J in R5
  • Surname: K-Å in G1

  Students with special exam requirements must contact the department office at least three working days before the exam.

  Telephone: 735 93520

  Email: mat.fag@math.ntnu.no

  If you have a conflict with the time you must let me know as soon as possible.

  The syllabus consists of the material up to and including week 40.

  Solutions now available as PDF.

  Grades are now available via student number.

  Student Id:

• Final.

  December 6th, 9am, 4hrs.

  Solution now available in PDF.

  Note: these solutions are not necessarily the only way to do the problems, and will tend to be longer than was actually required to get the marks since they are intended to help you (and future students) understand the answers.

The best preparation for the exams is to do the problem sets.

Final Grade

The rules for obtaining a final grade are as follows:

1. You must take the final exam to get a final grade on this course.

2. To be allowed to take the final exam, you must have passed eight of the homeworks.

   If you have taken this course before and have been allowed to take the final exam before then this still stands: you will be allowed to take the final exam again regardless of what assignments you do for the course this time around.

3. The midterm only counts positively and if it counts it counts for 20%. That is, your actual grade is max(.8F + .2M, F) where F is your final grade and M your midterm grade.

   Note that your actual grades on the homeworks do not enter into this calculation. They merely qualify you to take the exam.

   That the midterm only counts positively is new for this year. You can confirm this on the official description linked to here

Main Page

These documents are now under a mild lockdown. If you think you should be able to access them and can't, contact me.

1. Homework assignment one as a PDF.

   The material in the first few lectures is not in any of the course materials so here is a brief summary. Possible further sources of information include the Wikipedia articles on Naive Set Theory and Zermelo Set Theory, and the book Naive Set Theory by Paul Halmos.

   Solution now available as a PDF.

2. Homework assignment two as a PDF.

   Due in before Friday September 5th at 15:00.

   You will need access to a copy of the Kreyszig notes for one of the questions.

   Solution now available as a PDF.

3. Homework assignment three as a PDF.

   Due in before Friday September 12th at 15:00.

   You will need access to a copy of the Kreyszig notes for one of the questions. I have prepared a handout on some extras from metric space theory that are not covered in the Kreyszig notes; currently, it includes a definition of a subsequence and the list of all the types of function between metric spaces that were done in class.

   Solution now available as a PDF.

4. Homework assignment four as a PDF.

   Due in before Friday September 19th at 15:00.

   Solution now available as a PDF.

5. Homework assignment five as a PDF.

   Due in before Friday September 26th at 15:00.

   Solution now available as a PDF.

6. Homework assignment six as a PDF.

   Due in before Friday October 3rd at 15:00.

   Solution now available as a PDF.

7. No homework in week 41 due to the midterm in week 42.

8. Short homework due to the midterm this week. Homework assignment seven as a PDF.

   Due in before Friday October 17th at 15:00.

   Solution now available as a PDF.

9. Homework assignment eight as a PDF.

   Due in before Friday October 24th at 15:00.

   Solution now available as a PDF.

10. Homework assignment nine as a PDF.

    Due in before Friday October 31st at 15:00.

    Solution now available as a PDF.

11. Homework assignment ten as a PDF.

    Due in before Friday November 7th at 15:00.

    Solution now available as a PDF.

12. Homework assignment eleven as a PDF.

    Due in before Friday November 14th at 15:00.

    Solution now available as a PDF.

13. Homework assignment twelve as a PDF.

    Due in before Friday November 21st at 15:00.

    Solution now available as a PDF.

    Note: the number of qualifying assignments can be checked using the form in the assessment section.

Main Page

As an aid to following this course, I shall put up a basic scheme of work here. Although (almost) all the material can be found in the texts, we are not always working through it in the same order so this is intended as an aid to finding the relevant part of the notes. To that end, it will not be a detailed scheme but just the references to the course texts (and other notes).

1. 19th August. Introduction and Naive Set Theory. Ref: Notes on Set Theory.

2. 20th August. Continuing on Naive Set Theory (axioms). Ref: Notes on Set Theory.

3. 26th August. Continuing on Naive Set Theory (functions). Ref: Notes on Set Theory.

4. 27th August. Metric Spaces. Ref: Kreyszig 1.1.

5. 2nd September. Convergence. Ref: Kreyszig 1.4.

6. 3rd September. Functions of Metric Spaces. Ref: Kreyszig 1.3 (only part of this); Extras on Metric Spaces.

7. 9th September. Notions of equivalence; neighbourhoods and continuity-at-a-point in terms of neighbourhoods. Ref: Kreyszig 1.3.

8. 10th September. Open sets; contraction maps and fixed points. Ref: Kreyszig 1.3, 5.1.

9. 16th September. Contraction mapping theorem, examples. Ref: Kreyszig 5.1, 5.2, 5.3.

10. 17th September. Introduction to Linear Algebra. Example of polynomials of degree at most k.

11. 23rd September. Gaussian Elimination. Ref: Strang ch 1.

12. 24th September. Abstract vector spaces. Ref: Strang 2.1.

13. 30th September. Linear maps. Ref: Strang 2.6.

14. 1st October. Dimension and bases. Ref: Strang 2.3.

    Midterm syllabus is everything above this line.

    ---

15. 7th October. Representing subspaces (extension of bases). Ref: Strang 2.3.

16. 8th October. Representing surjections (reduction of spanning sets), kernel and image, representing linear transformations. Ref: Strang 2.

17. 14th October. Representing linear transformations. The Rank-Nullity Theorem. Ref: Strang 2.

18. 15th October. Using Gauss Elimination to find bases. Ref: Strang 2.4

19. 21st October. The LU factorisation. Ref: Strang 1.5

20. 22nd October. The minimum polynomial; (real) Jordan canonical form; generalised eigenvalues. Ref: Strang 5 (note that Strang introduces complex matrices in 5.5 and so 5.6 (Jordan form) is for complex matrices); Down with Determinants.

21. 28th October. What is an angle? Normed vector spaces and the parallelogram law. Ref: Young ch 2.

22. 29th October. Inner product spaces. Ref: Young ch 1.

23. 4th November. Cauchy-Schwarz inequality. Closest point property for Hilbert spaces. Ref: Young chs 1 and 3.

24. 5th November. Orthonormal bases. Ref: Young ch 4.

25. 11th November. Gram-Schmidt and least-squares. Ref: Strang ch 3 (3.3 and 3.4)

26. 12th November. Adjoints.

27. 18th November. Normal operators. Ref: Strang ch 6

28. 19th November. Self-adjoint operators. Ref: Strang ch 6

Main Page

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 to 2hrs and relocated:

  MA24 (Junge) and G21 (Fonn): 16:15 - 18:00 (starting week 35).

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 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:

• Kristoffer Hellton hellton@stud.ntnu.no
• Minxian Le minxian@stud.ntnu.no

Main Page

Here are some of last years' homework assignments and their solutions and some old midterms with solutions.

Assignments

• 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.

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.

Main Page

As a postscript to this course, here is a strange poem I came across recently. I've been wondering what a reasonable translation into Norwegian would look like.

I've heard it's been said, by and by,
(But have never yet tried asking why)
That you'd be a mug
To fashion a rug
From so strange a material as rye.

More Messages