Kursussøgning, efter- og videreuddannelse – Københavns Universitet

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Kursussøgning, efter- og videreuddannelse

Advanced Vector Spaces (AdVec)

Practical information
Study year 2016/2017
Block 1
Programme level Full Degree Master
Course responsible
  • Henrik Schlichtkrull (8-7e6e7377746e737f4b786c7f73397680396f76)
  • Department of Mathematical Sciences
Course number: NMAK15005U

Course content

This course covers the fundamentals of linear and multilinear algebra as well as more advanced subjects within the field, from a theoretical point of view with emphasis on proofs.

Topics include

1. Fundamentals of finite dimensional vector spaces over a field
2. Duality
3. Direct sums and quotient spaces
4. Multilinear algebra. Tensors and alternating tensors
5. Determinant and trace. The Cayley-Hamilton theorem
6. Generalized eigenspaces and the Jordan normal form
7. Real and complex Euclidean structure
8. Spectral theory of commuting normal operators
9. Positive operators, polar decomposition and calculus of operators
10.Quadratic forms and conics

Learning outcome

Knowledge: Central definitions and theorems from the subjects mentioned in the description of contents.

Skills: To follow and reproduce logical reasoning at an abstract level within the subjects mentioned in the description of contents. To give an oral presentation of a mathematical subject.

Competencies: To apply abstract results from the curriculum to concrete problems

Recommended prerequisites

Basic group theory and linear algebra, as covered by the courses LinAlg and Alg1 or equivalent.

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MSc Programme in Mathematics
MSc Programme in Statistics


Study Board of Mathematics and Computer Science

Course type

Single subject courses (day)


1 block


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Teaching and learning methods

5 hours of lectures and 4 hours of exercises per week for 7 weeks


no limit




Category Hours
Lectures 35
Theory exercises 28
Preparation 142
Exam 1
English 206


Type of assessment

Oral examination, 30 min
Oral examination with 30 minutes of preparation before the exam


All aids allowed during the preparation time. No aids allowed for the examination.

Marking scale

7-point grading scale

Criteria for exam assessment

The student must in a satisfactory way demonstrate that he/she has mastered the learning outcome of the course.

Censorship form

External censorship


Oral examination, 30 minutes plus 30 minutes of preparation before the exam. If the mandatory assignment has not been approved before the regular exam, the oral re-exam will include problem solving. In this case the timeslots for exam and preparation will each be prolonged by 15 minutes, and the grading will be based on the total performance.

In either case, all aids are allowed during the preparation time, but no aids allowed during the examination. 

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