# TMA4145 - Linear Methods

### Examination arrangement

Examination arrangement: School exam

Evaluation Weighting Duration Grade deviation Examination aids
School exam 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 the course is attended by students who don't master a Scandinavian language.

• Exercises

### Further on evaluation

Retake of examination may be given as an oral examination.

### Course materials

Will be announced at the start of the course.

### Credit reductions

Course code Reduction From To
SIF5020 7.5
More on the course
Facts

Version: 1
Credits:  7.5 SP
Study level: Third-year courses, level III

Coursework

Term no.: 1
Teaching semester:  AUTUMN 2023

Language of instruction: English, Norwegian

Location: Trondheim

Subject area(s)
• Mathematics
• Technological subjects
Contact information
Course coordinator:

Department of Mathematical Sciences

# Examination

#### Examination arrangement: School exam

Term Status code Evaluation Weighting Examination aids Date Time Examination system
Autumn ORD School exam 100/100 2023-12-04 09:00
Summer UTS School exam 100/100
• * 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.
Examination

For more information regarding registration for examination and examination procedures, see "Innsida - Exams"

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