NMAK15005U  Advanced Vector Spaces (AdVec)

Volume 2018/2019

MSc Programme in Mathematics
MSc Programme in Statistics


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

Basic group theory and linear algebra, as covered by the courses LinAlg and Alg1 or equivalent.
5 hours of lectures and 4 hours of exercises per week for 7 weeks
7,5 ECTS
Type of assessment
Oral examination, 30 minutes
Oral examination with 30 minutes of preparation before the exam
Exam registration requirements

A mandatory assignment must be approved before the exam.

Only certain aids allowed

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

Marking scale
7-point grading scale
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 student will be given a written assignment three weeks before the re-exam week. The assignment must be approved no later than two weeks before the exam.

All aids are allowed during the preparation time, but no aids allowed during the examination. 

Criteria for exam assesment

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

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