MAT4000 – Numbers, spaces and linearity
Course description
Course content
Euclid's algorithm, prime-power factorization, congruence, Fermat's little theorem, Euler's theorem, Wilson's theorem, quadratic residues, sums of squares, distribution of primes. Vector spaces and linear transformations, matrix representations and change of basis, inner product spaces, spectral theory in finite dimensional spaces, Schur triangulation, the Cayley-Hamilton theorem, an introduction to Jordan normal form. Some applications, covering from cryptography, geometry (projections, reflections, rotations) to analysis (differential equations and discrete Fourier analysis) are also included.
Learning outcome
You are first introduced to classical number theory and learn how it plays a central role in public key cryptography. Thereafter, linear algebra, which you have already met in introductory courses, is developed in a broader context (by considering vector spaces over a field, with emphasis on the real and complex cases). The aim is to provide you with a thorough understanding of the concepts and the main results of linear algebra, which are of fundamental importance for most branches of modern mathematics. The theory is illustrated with several useful applications.
Admission
Students who are admitted to study programmes at UiO must each semester register which courses and exams they wish to sign up for in Studentweb.
If you are not already enrolled as a student at UiO, please see our information about admission requirements and procedures.
Prerequisites
Recommended previous knowledge
The course follows on from MAT1100 – Calculus, MAT1110 – Calculus and Linear Algebra and MAT1120 – Linear Algebra. It will be useful to have taken MAT2200 – Groups, Rings and Fields, but this is not a prerequisite for the course.
Overlapping courses
- 10 credits overlap with MAT3000 – Numbers, spaces and linearity (discontinued)
- 3 credits overlap with MA115
- 3 credits overlap with MA215
The information about overlaps is not complete. Contact the department for more information if necessary.
Teaching
4 hours of lectures/exercises every week for the duration of the semester.
Examination
Two mandatory assignments that must be passed within given deadlines in order to be allowed to take the final exam. The compulsory assignments must be done with presentation software for mathematics(Latex). The objective is that the student becomes familiar with and master electronic tools for creating written mathematics and be able to present his/her own mathematical work in an electronic format.
Final mark based on written examination at the end of the semester.
Examination support material
No examination support material is allowed.
Language of examination
Subjects taught in English will only offer the exam paper in English.
You may write your examination paper in Norwegian, Swedish, Danish or English.
Grading scale
Grades are awarded on a scale from A to F, where A is the best grade and F is a fail. Read more about the grading system.
Explanations and appeals
Resit an examination
This subject offers new examination in the beginning of the subsequent term for candidates who withdraw during an ordinary examination or fail an ordinary examination. Deferred examinations for students who due to illness or other valid reason of absence were unable to sit for their final exams will be arranged at the same time. (These valid reasons has to be documented within given deadlines.)
For general information about new and deferred examination, see
/studier/admin/eksamen/sykdom-utsatt/mn/index.html
Withdrawal from an examination
It is possible to take the exam up to 3 times. If you withdraw from the exam after the deadline or during the exam, this will be counted as an examination attempt.
Special examination arrangements
Application form, deadline and requirements for special examination arrangements.
Evaluation
The course is subject to continuous evaluation. At regular intervals we also ask students to participate in a more comprehensive evaluation.