TMA4145 - Linear Methods


Examination arrangement

Examination arrangement: Written examination
Grade: Letters

Evaluation form Weighting Duration Examination aids Grade deviation
Written examination 100/100 4 hours D

Course content

Linear and normed spaces. Completeness, Banach spaces and Banach's fixed point theorem. Picard's theorem. Linear transformations. Inner product spaces, projections, and Hilbert spaces. Orthogonal sequences and approximations. Linear functionals, dual space, and Riesz' representation theorem. Spectral theorem, Jordan canonical form, and matrix decompositions.

Learning outcome

1. Knowledge. The student has knowledge of central concepts in the theory of vector spaces, normed spaces and Hilbert spaces. In the theory of vector spaces the main objective is that the student understand the transition from Euclidean spaces to general vector spaces. This includes an understanding of isomorphisms and bases of finite dimensional vector spaces and the relationship between linear transformations and matrices. The student is familiar with principles of matrix factorization.In the theory of normed spaces a key objective is that the student understand the Banach fixed point theorem. This includes an understanding of convergence of sequences and continuous functions. The student masters the basic concepts from the theory of Hilbert spaces, including orthogonality, closest point and duality. The student understands the Riesz representation theorem.

2. Skills. The student is able to apply his or her knowledge of the theory of vector spaces, normed spaces and Hilbert spaces to solve concrete problems. A key skill is that the student is able to combine results and construct new proofs using the theory acquired in the course.

Learning methods and activities

Lectures and mandatory exercises. The lectures will be given in English if they are attended by students from the Master's Programme in Mathematics for International students. Retake of examination may be given as an oral examination.

Compulsory assignments

  • Exercises

Further on evaluation

see «Teaching methods and activities».

Specific conditions

Exam registration requires that class registration is approved in the same semester. Compulsory activities from previous semester may be approved by the department.

Course materials

Will be announced at the start of the course.

Credit reductions

Course code Reduction From To
SIF5020 7.5



Examination arrangement: Written examination

Term Statuskode Evaluation form Weighting Examination aids Date Time Room *
Autumn ORD Written examination 100/100 D
  • * The location (room) for a written examination is published 3 days before examination date.
If more than one room is listed, you will find your room at Studentweb.