It is important to understand how mathematical reasoning works. For this reason students are asked to learn definitions at home and derive proofs for the theorems together in class. So we will use rotation mode where in each group of students a couple of proofs are discussed in details. Students at home can participate via the Webconference tool MS Teams.
Students have to upload their homework solutions using the myLearn platform (i.e., this web site). This also has to be done by students who participate in the class room lectures.
Students which cannot participate in the class room lectures may be asked to present their homework solutions using a web conference tool like MS Teams. Students who participate in the class room lectures may have to present their solutions on the white board.
Students who cannot participate in the class room lectures can do the intermediate tests and the final exam online at the same time as the class room exams. Details for these tests will be announce in time.
Working with mathematical models requires two skills: First one needs to be familiar with techniques for handling terms and formulae and with methods for solving particular problems like finding extrema of a given function. Learning and applying such procedures is already part of the course "Mathematische Methoden". The second skill is the investigation of structural properties of a given model. One has to find conclusions that can be drawn from one's model and find convincing arguments for these.
In this course our emphasis is on mathematical reasoning. New notions are declared in definitions. Conclusions are stated in theorems. Proofs demonstrate that our claims hold in all cases where the given conditions are satisfied. Counter examples may show that a conjecture is wrong. Examples help us to deal with often abstract concepts.
We learn these ideas in the framework of linear algebra. We do this for several reasons. It provides the mathematics for all linear models which are important in, e.g., econometric studies. Moreover, in mathematics non-linear functions are often replaced by appropriate linear ones in order to make a problem tractable. The concepts in linear algebra are abstract but we often can use examples from our three dimensional world to illustrate these. Moreover, few definitions give way to rich structure with comparatively short proofs.
In summary, the course has the following topics:
- Fundamental of mathematical reasoning
- Definition, theorem, proof, necessary condition, sufficient condition
- Proof techniques
- Vector space, basis, dimension
- Linear transformation and matrix
- Distance, norm and Euclidean space
- Eigenvalues and eigenvectors
Students of this course understand fundamental principles that are indispensable for understanding higher mathematics. This includes
- the concepts of definitions, theorems and proof
- the important distinction between necessary and sufficient conditions
- mathematical reasoning
- proof techniques: direct and indirect proof, proof by contradiction
In addition they understand the basic concepts of linear algebra and can apply these to problems that occur in the analysis of linear models. A typical application are formulae used in econometrics.
For this lecture participation is obligatory. Students are allowed to miss a maximum of 20% (no matter if excused or not excused).
It is important that students learn to find their own approach to mathematics. Thus the course is organized in the following way. In each unit new concepts are presented. This includes definitions as well as some properties with prototypical proofs. In their homeworks students work on problems related to these new concepts and thus acquire both more knowledge about the mathematical concepts as well as expertise in mathematical reasoning. Ideally the students then get feedback about their solutions. Selected students will then present their solutions in the class.
Competent handling of terms, formulae, equalities and inequalities is a necessary prerequisite to master this course.