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Linear Algebra II (MCS)

The exams have been corrected. Marks appear in the show case next to Room 209 in Mathebau. You will have the opportunity to have a look at your marked exam ("Klausureinsicht") on Friday the 28th of September at 10:00 in Room 201 in Mathebau.

The exam will take place on Tuesday the 18th of September from 9:00-13:00 in S202-C205 (and not in S206-030, as originally announced). You are allowed to bring books, the lecture notes, your notes, the exercise sheets and their solutions to the exam, but no electronic devices.

In the final week the lecture will not take place on Monday the 16th of July, but on Wednesday the 18th of July at 11:40-13:30 in S1|03/23.

To compensate for Pentecoste, there will be an additional lecture on Wednesday May the 30th at 9:50-11:40 in S1|03/23.

Regrettably, there was a mistake in exercise E6.1. The last entry in the second row of the matrix A should have been -2 instead of 2.

Because of Ascension and Corpus Christi, the exercise classes in weeks 20 and 23 will be on Tuesdays. So there will be exercise classes on May the 15th and June the 5th, from 13:30-15:10 in S1|02/34.

Please be present at the exercise class on Thursday April 19 so that you can sign up for the exercise classes.

Lecture notes are available at the link below.

First lecture on the 16th of April and first exercise class on the 19th of April.

NameOfficeadditional contact information
Prof. Dr. Martin OttoS2|15 / 207
Dr. Benno van den BergS2|15 / 203LZM Friday 11:35-12:30

Course notes are available here; also an erratum sheet correcting a mistake in section 1.6.1 (page 42).
Linear algebra is one of the fundamental areas of mathematics. Together with calculus it forms one of the cornerstones in first year undergraduate education in mathematics.

The core topic of linear algebra is the investigation of vector spaces and linear maps. Linear algebra has close links with geometry and with various application areas inside mathematics and beyond.

Techniques and methods from linear algebra are ubiquitous in many branches of pure and applied mathematics, as well as for instance in physics and engineering, in computer graphics, or in information theory.

This course, Linear Algebra II, is a continuation on the basis of Linear Algebra I, with more advanced material and wider scope.

The main focus in subject matter is on the following three topics:

  • The investigations into eigenvalues/eigenvectors and the representation of endomorphisms in suitably chosen bases over finite dimensional vector spaces, including Jordan normal form.
  • Real and complex vector spaces with additional "metric" structure: Euclidean and unitary spaces.
  • The study of bilinear and quadratic forms.
  • TimePlace
    Monday, 9.50-11.30 am S2|04 / 213

    The course is taught in English. Exercise groups from an integral part of the course. Students are strongly encouraged to participate actively in the exercises and to hand in weekly homework assignments. These components of the course are essential not only for understanding the material taught, but in order to experience and practice the core mathematical activities of problem analysis, problem solving and rigorous presentation of mathematical thought.

    Group work during exercises, with guidance by the demonstrator, as well as feedback on written solutions to the assignments are particularly important parts of the learning experience, and as essential for satisfactory performance on the course as the actual lectures. Students are also encouraged to ask questions during the lectures and contact hours. Course notes, as well as exercise sheets and related material are being made available electronically, via the links provided on this page.

    GroupTimePlaceTutoroffice hours
    ExercisesThursday, 16.15 - 17.55 pmS1|03 / 104 Benno van den BergWe 14:00-15:00

    Exercise sheets are provided both for the Thursday exercise groups. The sheet for the exercise group will typically comprise five or six exercises. The first few of these are primarily intended for group work during the exercises, with the remainder serving as homework assignments to be submitted on the following Monday and discussed the Thursday after.

    It is extremely important that students train their skills in writing up solutions and formulating mathematical thought and argument (also for the exam, but by no means only for that reason). Written solutions to any of the Thursday exercises, including those covered in group work, can be handed in as homework. These will be marked and returned with feedback.

    Participation will be monitored, including performance in homework submissions; successful participation/performance will be certified with an "Übungsschein" (which, although not a formal requirement in the course, may be counted as an additional bonus).

    Exercise Sheets
    1 2 3 4 5 6 7 8 9 10 11 12 13 14
    Exercise Sheet Solutions
    1 2 3 4 5 6 7 8 9 10 11 12 13 14
    Note that the required exam for MCS bachelor students is the module exam covering Linear Algebra I and II, primarily offered in September.
    There are plenty of textbooks both in German and in English on the topic of linear algebra, at various levels but also differing widely with respect to comprehensiveness and structure. Students are encouraged to explore which books suit their tastes. Most relevant books are available in the mathematics library, which has a shelf in the reading room dedicated to linear algebra and one with English books especially for the MCS students. (There is also a section with mathematical dictionaries.)

    Some suggestions (from the library):

    Anton: Elementary Linear Algebra, 7th edition, Wiley also in German translation, Spektrum Verlag.

    Artmann: Lineare Algebra, Birkhäuser

    Beutelsbacher: Lineare Algebra, Vieweg

    Brieskorn: Lineare Algebra und Analytische Geometrie (I,II), Vieweg

    Bröcker: Lineare Algebra und Analytische Geometrie, Birkhäuser

    Curtis: Linear Algebra - An Introductory Approach, Springer

    Fischer: Lineare Algebra, 11. Aufl., Vieweg

    Greub: Linear Algebra, Springer

    there is also a German edition

    Jänich: Lineare Algebra, 10. Aufl., Springer also in English translation, Springer

    Kaye, Wilson: Linear Algebra, Oxford University Press

    Klingenberg, Klein: Lineare Algebra und Analytische Geometrie, BI

    Koecher: Lineare Algebra und Analytische Geometrie, Springer

    Kwak, Hong: Linear Algebra, Birkhäuser

    Lingenberg: Lineare Algebra, BI

    Strang: Linear Algebra and its Applications, Academic Press