Short description

Students are familiarised with vector spaces, linear mappings, eigenvalues and eigenvectors. They learn how to mathematically describe linear mappings on vector spaces, using vectors and matrices, and apply these concepts to Fourier analysis and to the solution of linear differential equations.

Module coordinator

Schmid Matthias (scmi) 
Learning objectives (competencies)

Objectives 
Competences 
Taxonomy levels 
You are familiar with the abstract notion of a vector space and its
subspaces. You can describe vectors as coordinate vectors with
respect to some basis. In particular, you know the Fourier series
as an application of this concept.

F, M 
K2, K3 
You are familiar with linear mappings between vector spaces and you are
able to describe them with respect to arbitrary bases using matrices and
vectors.

F, M 
K2, K3 
You can calculate eigenvalues and eigenvectors of linear mappings
and examine matrices for their diagonalizability. You are able to apply the
diagonalization of matrices as an important practical insight from linear algebra
to technical contexts.

F, M 
K2, K3 
You are able to identify and to solve linear ordinary differential
equations with constant coefficients.

F, M 
K2, K3 

Module contents

Vector Spaces

Vector spaces and vector space axioms

Subspaces
 Linear independence of vectors
 Basis and dimension of vector spaces

Inner product, norm and orthonormal bases of vector spaces

Fourier Series
Linear Mappings

Linear Mappings

Examples of linear mappings (reflections, scalings, rotations and projections)

Fundamental spaces of a matrix (null space and column space)

Invertible linear mappings (isomorphisms)

Change of basis of a vector space
Eigenvalues and Eigenvectors

Calculation of eigenvalues and eigenvectors

Basis of eigenvectors and diagonalization of matrices

Applications of diagonalization (e.g. linear ordinary differential equations)

Teaching materials

Depending on the lecturer 
Supplementary literature


Lernbuch Lineare Algebra und Analytische Geometrie,
Gerd Fischer, Florian Quiring,
Springer Spektrum Verlag, 2. Auflage
http://dx.doi.org/10.1007/9783834823793

Lineare Algebra für Naturwissenschaftler und Ingenieure,
Michael Ruhrländer,
Pearson Studium
ISBN 9783868942712

Formeln, Tabellen, Begriffe (Mathematik, Physik, Chemie),
Orell Füssli Verlag,
ISBN 9783280040294

Prerequisites


Teaching language

(X) German ( ) English 
Part of International Profile

( ) Yes (X) No 
Module structure

Type 2b 

For more details please click on this link: T_CL_Modulauspraegungen_SM2025 
Exams

Description 
Type 
Form 
Scope 
Grade 
Weighting 
Graded assignments during teaching semester 
In consultation 
written or orally 

Grade 
20% 
Endofsemester exam 
Exam 
written 
120 min 
Grade 
80% 

Remarks


Legal basis

The module description is part of the legal basis in addition to the general academic regulations. It is binding. During the first week of the semester a written and communicated supplement can specify the module description in more detail. 