2018/2019

## Research Seminar "Advanced Linear Algebra"

Type:
Optional course (faculty)

Delivered by:
Faculty of Mathematics

When:
3, 4 module

Instructors:
Karine Kuyumzhiyan

Language:
English

ECTS credits:
5

### Course Syllabus

#### Abstract

This course is aimed to introduce main notions of Linear Algebra and their instances in other fields of mathematics.Abstraction refers to the setting of general vector spaces, with finite dimension or not, with given basis or not, over an arbitrary field. Approximation asks for best answers to linear systems when exact solutions either don't exist or are not worth computing to arbitrary precision. This is related to variation of subspaces or of entries of matrices: what kind of geometric space is the set of all k-dimensional subspaces? And what happens to the eigenvalues of a matrix when the entries of the matrix are wiggled? Positivity refers to the entries of a matrix or to the eigenvalues of a symmetric matrix; both have interesting, useful consequences. Convexity stems from the observation that a real hyperplane H splits a real vector space into two regions, one on either side of H. Intersections of regions like this yield familiar objects like cubes, pyramids, balls, and eggs, the geometry of which is fundamental to many applications of linear algebra. Throughout the course, motivation comes from many sources: statistics, computer science, economics, and biology, as well as other parts of mathematics. We will explore these applications, particularly in projects (paper plus oral presentation) on topics of the students' choosing. Students will continue to develop their skills in mathematical exposition, both written and oral, including proofs. Pre-requisites : some notions from the Fall term course «Algebra and Arithmetics» will be used, especially, fields and groups.

#### Expected Learning Outcomes

- Can rigorously apply linearization techniques to solving problems in various parts of mathematics as well as physical sciences

#### Course Contents

- Matrices and Matrix Operations. Systems of Linear Equations. Cramer's Rule
- Vector spaces
- Linear Transformations
- Symmetry and Permutation Representations
- Bilinear Forms
- Linear Groups
- Basics of Representation Theory

#### Bibliography

#### Recommended Core Bibliography

- Körner, T. W. (2013). Vectors, Pure and Applied : A General Introduction to Linear Algebra. Cambridge: Cambridge University Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=508905

#### Recommended Additional Bibliography

- Anthony, M., & Harvey, M. (2012). Linear Algebra : Concepts and Methods. Cambridge, UK: Cambridge eText. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=443759
- Anton, H. (2014). Elementary Linear Algebra : Applications Version (Vol. 11th edition). Hoboken, NJ: Wiley. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1639248
- Ford, W. (2015). Numerical Linear Algebra with Applications : Using MATLAB (Vol. First edition). London: Academic Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=485990
- Fuad Aleskerov, Hasan Ersel, & Dmitri Piontkovski. (2011). Linear Algebra for Economists. Springer. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsrep&AN=edsrep.b.spr.sptbec.978.3.642.20570.5
- Shores, T. S. Applied linear algebra and matrix analysis. – New York : Springer, 2007.