Course - Linear Methods - TMA4145
TMA4145 - Linear Methods
About
Examination arrangement
Examination arrangement: Written examination
Grade: Letters
| Evaluation | Weighting | Duration | Grade deviation | Examination aids |
|---|---|---|---|---|
| Skriftlig | 100/100 | 4 timer |
Course content
Metric spaces, completeness and the contraction principle. Picard's theorem and Jacobi iteration. Recapitulation of linear algebra. Inner product spaces, projections and least squares problems. The spectral theorem and Jordan decomposition. The Cayley-Hamilton theorem. Positive definite matices, pseudo inverse and singular value decomposition. Banach spaces and Hilbert spaces. Orthogonal systems and approximations. Linear functionals, dual spaces, and the Riesz representaion theorem in Hilbert spaces.
Learning outcome
1. Knowledge. The student has knowledge of central concepts in the theory of metric spaces, vector spaces and Hilbert spaces. In the theory of metric spaces a key objective is that the student understand the Banach fixed point theorem. This includes an understanding of metric spaces, convergence of sequences and continuous functions. 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. 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 metric spaces, vector 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
Recommended previous knowledge
The course is based on TMA4100/05/15/20 Calculus 1/2/3/4K, or equivalent. MA1101 Basic Calculus I, MA1102 Basic Calculus II, MA1103 Vector Calculus, MA1201 Linear Algebra and Geometry, MA1202 Linear Algebra with Applications, MA2105 Complex Function Theory with Differential Equations, or equivalent.
Course materials
Will be announced at the start of the course.
Credit reductions
| Course code | Reduction | From | To |
|---|---|---|---|
| SIF5020 | 7.5 |
Version: 1
Credits:
7.5 SP
Study level: Third-year courses, level III
Term no.: 1
Teaching semester: AUTUMN 2013
Language of instruction: English, Norwegian
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- Mathematics
- Technological subjects
Department with academic responsibility
Department of Mathematical Sciences
Examination
Examination arrangement: Written examination
- Term Status code Evaluation Weighting Examination aids Date Time Examination system Room *
- Autumn ORD Skriftlig 100/100 2013-12-11 15:00
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Room Building Number of candidates - Summer KONT Skriftlig 100/100 2014-08-07 09:00
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Room Building Number of candidates
- * 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.
For more information regarding registration for examination and examination procedures, see "Innsida - Exams"