(Created 2009-08-11.)
 MATHEMATICS - SYSTEMS AND TRANSFORMS FMAF05

Higher education credits: 7. Grading scale: TH. Level: G2 (First level). Language of instruction: The course will be given in Swedish. FMAF05 overlap following cours/es: FMA450, FMA030, FMA036, FMA062, FMA450, FMAF10, FMA030, FMA036, FMA062, FMA450 och FMAF10. Compulsory for: E2, F2, I2, Pi2. Alternative for: D2. Optional for: C4, N3. Course coordinator: Director of Studies, Lars-Christer Böiers, Matematik. Recommended prerequisits: FMAF01 Analytic functions. Assessment: Written test comprising theory and problem solving. Computer work. Some written assignements should be completed before the exam. Home page: http://www.maths.lth.se/matematiklth/vitahyllan/vitahyllan.html.

Aim
The aim of the course is to present mathematical concepts and methods from linear algebra and analysis which are important in systems theory, continuous and discrete, and for further studies within e.g. mathematics, economy, physics, mathematical statistics, mechanics, control theory, signal theory and for future professional work. The aim is also to develop the student's ability to solve problems, to assimilate mathematical text and to communicate mathematics.

Knowledge and understanding
For a passing grade the student must

be familiar with the significance of eigenvalues in the context of stability and resonance, in continuous as well as discrete systems.

be able to describe and use the concepts of linearity, time and space invariance, stability, causality, impulse response and transfer function, in continuous as well as discrete time.

be able to describe the structure of an exponential matrix, and be able to compute exponential matrices in simple cases.

be able to characterize different types of quadratic forms using eigenvalue methods and via a completion of squares.

be able to define the concept of convolution, continuous and discrete, and to use convolutions in the context of systems and in the description of certain types of integral equations.

have some experience and understanding of mathematical and numerical software.

Skills and abilities
For a passing grade the student must

be able to show capability independently to choose appropriate methods to solve systems of linear differential and difference equations, and to carry out the solution essentially correct.

be able to show capability to use eigenvalue techniques, elementary distribution theory, function theory, Fourier and Laplace transforms and convolutions in problem solving within the theory of linear systems.

in connection with problem solving be able to show capability to integrate knowledge from the different parts of the course.

with proper terminology, in a well structured way and with clear logic be able to explain the solution to mathematical problems in the course.

Contents
Linear algebra: Spectral theory, quadratic forms.

Systems of linear differential equations: Equations in state form. Solution via diagonalization. Stability. Stationary solutions and transients. Solution via exponential matrix.

Input/output relations: Linearity, time and space invariance, stability, causality. Convolutions. Elementary distribution theory. Transfer and frequency function. Discrete systems.

Fourier analysis: The Laplace and Fourier transforms. Inversion formulae, the convolution theorem and Parseval's formula. Transform theory and analytic functions. Applications in differential equations and systems of differential equations.

Literature
Spanne, S: Lineära system. KF-Sigma 1997.