NMAK15005U Advanced Vector Spaces (AdVec)

Volume 2021/2022

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.

Subjects include

1. Fundamentals of finite dimensional vector spaces over a field
2. Linear maps and dual space
3. Bilinear forms and quadratic forms
4. Direct sums, quotient spaces and tensor products
5. Eigenvectors and spectral decompositions
6. Generalized eigenspaces and the Jordan normal form
7. Real and complex Euclidean structure
8. Spectral theory of normal operators
9. Normed spaces, Hilbert spaces and bounded operators
10. Perron-Frobenius theorm
11. Multilinear algebra and determinants
12. Factorizations of matrices

Learning Outcome

Knowledge: Central definitions and theorems from the subjects mentioned in the description of contents. In particular, the following notions are considered central:

Linear dependence, basis, dimension, quotient space, quotient map, invariant subspace, rank, nullity, dual space, dual basis, adjoint map, direct sum, projection, idempotent map, bilinear form, alternating form, quadratic form, positive definite form, non-degenerate, tensor product, multilinear form, wedge product, determinant, trace, eigenvalue, eigenvector, eigenspace, spectrum, spectral radius, geometric multiplicity, algebraic multiplicity, characteristic polynomial, diagonability, flag, inner product, Hilbert space, self-adjoint map, normal map, unitary map, nilpotent map, cyclic vector, generalized eigenspace, operator norm, spectral radius, positive definite map, principal minors, leading principal minors.



To follow and reproduce proofs of statements within the subjects mentioned in the description of contents and involving the notions mentioned above.

To understand the relationships between the different subjects of the course

To prepare and give a coherent oral presentation of a random mathematical topic within the curriculum of the course.

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

Academic qualifications equivalent to a BSc degree is recommended.
5 hours of lectures and 4 hours of exercises per week for 7 weeks
  • Category
  • Hours
  • Lectures
  • 35
  • Preparation
  • 142
  • Theory exercises
  • 28
  • Exam
  • 1
  • Total
  • 206
Continuous feedback during the course of the semester

Oral feedback will be given on students’ presentations in class

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 homework assignment of 5 days 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.

An approved mandatory assignment is valid for the re-exam in the year it was approved and for exam and re-exam the following year, but no longer.

If the mandatory assignment has not been approved before the regular exam, the student must contact the course coordinator when he/she registers for the re-exam. The student will then be given a written homework assignment of 5 days, at least four weeks before the re-exam week. The assignment must be approved no later than three 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.