# 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 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 in finite dimension
11. Perron-Frobenius theorem
12. Factorizations of matrices
Education

MSc Programme in Mathematics

MSc Programme in Statistics

MSc Programme in Mathematics with a minor subject

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, 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.

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 (30-minute preparation time)
Exam registration requirements

A mandatory homework assignment of 5 days must be approved 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
Re-exam

Oral examination, 30 minutes plus 30 minutes of preparation before the exam. All aids are allowed during the preparation time, but no aids allowed during the examination.

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 registering 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.

##### 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)

• 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

Placement
Block 1
Schedulegroup
A
Capacity
No limitation – unless you register in the late-registration period (BSc and MSc) or as a credit or single subject student.
Studyboard
Study Board of Mathematics and Computer Science
##### Contracting department
• Department of Mathematical Sciences
##### Contracting faculty
• Faculty of Science
##### Course Coordinators
• Henrik Schlichtkrull   (8-76666b6f6c666b77437064776b316e7831676e)
• Damian Longin Osajda   (2-737e4f7c7083773d7a843d737a)
Saved on the 14-02-2024

### Are you BA- or KA-student?

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Courseinformation of students