## Analysis and Linear Algebra

### Wintersemester 2015/16

Working with mathematical models requires two skills: First one needs to be familiar with techniques for handling terms and formulæ 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 Mathematik (Bakk.). 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. Counterexamples may show that a conjecture is wrong. Examples help us to deal with often abstract concepts.

In the first part of this course 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, part I of the course covers 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
• Projections
• Determinant
• Eigenvalues and eigenvectors
In the second part of the course we will learn the fundamental principles in analysis. The basic idea is to replace "complicated" functions locally by linear ones (i.e., their derivates) and apply our results from linear algebra. Of course we need a stringent definition of "replace locally". In summary, part II of the course covers the following topics:
• Sequences and Series
• Topology
• Derivatives
• Taylor Series
• Inverse and Implicit Function Theorem
• Convex Functions
• Static Optimization
• Integration
• Multiple Integrals