LINEAR ANALYSIS | FMA030 |

**Aim**

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

*Knowledge and understanding*

For a passing grade the student must

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

be familiar with the significance of eigenvalues in the context of stability and resonance.

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 and to use convolutions in the context of systems and in the description of certain types of integral equations.

*Skills and abilities*

For a passing grade the student must

be able to show capability to use eigenvalue techniques, elementary distribution 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***Sums and series*: Fourier series.

*Linear algebra*: Spectral theory, quadratic forms.

*Frequency analysis*: The Laplace and Fourier transforms, inversion formulae, the convolution theorem and Parseval's formula.

*Systems theory*: Linear operators, linear, causal and time invariant systems in input/output form in the time and frequency domains, impulse response, step response.

**Literature**

Spanne, S: Lineära system. KF-Sigma 1997. Supplementary material.