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. 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 and bounded operators
10. Perron-Frobenius theorem
11. Multilinear algebra and determinants
12. Factorizations of matrices
MSc Programme in Mathematics
MSc Programme in Statistics
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,
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 feedback will be given on students’ presentations in class
- ECTS
- 7,5 ECTS
- Type of assessment
-
Oral examination, 30 minutesOral 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
- Preparation
- 142
- Theory exercises
- 28
- Exam
- 1
- English
- 206
Kursusinformation
- Language
- English
- Course number
- NMAK15005U
- ECTS
- 7,5 ECTS
- Programme level
- Full Degree Master
- Duration
-
1 block
- Schedulegroup
-
A
- Capacity
- no limit
- Studyboard
- Study Board of Mathematics and Computer Science
Contracting department
- Department of Mathematical Sciences
Contracting faculty
- Faculty of Science
Course Coordinator
- Henrik Schlichtkrull (8-79696e726f696e7a4673677a6e34717b346a71)
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