Advanced Vector Spaces (AdVec)
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
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. Multilinear algebra and determinants
8. Real and complex Euclidean structure
9. Normed spaces, Hilbert spaces and bounded operators
10. Spectral theory of normal operators
11. Perron-Frobenius theorem
12. Factorizations of matrices
MSc Programme in Mathematics
MSc Programme in Statistics
MSc Programme in Mathematics with a minor subject
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, minimal polynomial, 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.
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 feedback will be given on students’ presentations in class
- 7,5 ECTS
- Type of assessment
Oral examination, 30 minutes
- Type of assessment details
- Oral examination with 30 minutes of preparation 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
Criteria for exam assessment
The student should convincingly and accurately demonstrate the knowledge, skills and competences described under Intended learning outcome.
Single subject courses (day)
- Theory exercises
- Course number
- 7,5 ECTS
- Programme level
- Full Degree Master
- Block 1
- no limit
The number of seats may be reduced in the late registration period
- Study Board of Mathematics and Computer Science
- Department of Mathematical Sciences
- Faculty of Science
- Henrik Schlichtkrull (8-77676c706d676c78447165786c326f7932686f)
- Damian Longin Osajda (2-737e4f7c7083773d7a843d737a)
Are you BA- or KA-student?
Courseinformation of students