NMAK15005U Advanced Vector Spaces (AdVec)
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.
1. Fundamentals of finite dimensional vector spaces over a field
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
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.
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.
Academic qualifications equivalent to a BSc degree is recommended.
Oral feedback will be given on students’ presentations in class
- 7,5 ECTS
- Type of assessment
- Oral examination, 30 minutesOral 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 must contact the course coordinator when he/she registers for the re-exam. The student will then be given a written assignment 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.
- Theory exercises