Analysis and Linear Algebra
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
- Eigenvalues and eigenvectors
- Sequences and Series
- Taylor Series
- Inverse and Implicit Function Theorem
- Convex Functions
- Static Optimization
- Multiple Integrals
- K. Houston: How to Think Like a Mathematician, Cambridge University Press, 2009.
- K. Sydsæter, P. Hammond, A. Seierstad, A. Strøm: Further Mathematics for Economic Analysis, Prentice Hall, 2005.
- K. Sydsæter, P. Hammond: Essential Mathematics for Economics Analysis, Prentice Hall, 3rd edition, 2008.
- A. C. Chiang, K. Wainwright: Fundamental Methods of Mathematical Economics, McGraw-Hill, New York, 2005
Last change: Mon Oct 7, 2015 by josef leydold