Advanced Vector Spaces (AdVec)

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.

Subjects include

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

Education

MSc Programme in Mathematics
MSc Programme in Statistics

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, geometric multiplicity, algebraic multiplicity, characteristic polynomial, diagonability, flag, inner product, 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.

 

Skills/Competencies:

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.

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

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.

Oral
Collective
Continuous feedback during the course of the semester

Oral feedback will be given on students’ presentations in class

ECTS
7,5 ECTS
Type of assessment
Oral examination, 30 minutes
Oral examination with 30 minutes of preparation before the exam
Aid
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
Criteria for exam assessment

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

Single subject courses (day)

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