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Bachelor 2021/2022

# Applied Methods of Linear Algebra

Category 'Best Course for New Knowledge and Skills'
Type: Elective course (Software Engineering)
Area of studies: Software Engineering
When: 3 year, 1, 2 module
Mode of studies: offline
Open to: students of one campus
Instructors: Dmitri Piontkovski
Language: English
ECTS credits: 5
Contact hours: 60

### Course Syllabus

#### Abstract

In the lecture course, we consider some topics of linear algebra beyond the standard first year course which are extremely important for applications. Mostly, these are applications to data analysis and machine learning, as well as to economics and statistics. We begin with inversions of rectangle matrices, that is, we discuss pseudo-inverse matrices (and their connections to the linear regression model). Among others, we discuss iteration methods (and their using in models of random walk on a graph applied to Internet search such as PageRank algorithm), matrix decompositions (such as SVD) and methods of dimension decreasing (with their connection to some image compression algorithms), and the theory of matrix norms and perturbation theory (for error estimates in matrix computations). The course includes also symbolic methods in systems of algebraic equations, approximation problems, Chebyshev polynomials, matrix functions such as exponents etc. We plan to invite some external lecturers who successfully apply linear algebra in their work. The students are also be invited to give their own talks on additional topics of applied or theoretical linear algebra.

#### Learning Objectives

• The aim the course is to provide both theoretical background and practical experience of solutions of linear algebra problems which appear in computer science, data analysis, mathematical modelling, machine learning, and economical models. The course covers some topics of matrix analysis and numerical methods of linear algebra as well as some elements of functional analysis and mathematical statistics. We provide a number of useful algorithms which can be implemented and used by students. A number of of these algorithms are in the core of modern machine learning and data analysis.

#### Expected Learning Outcomes

• Upon completion of this course students would be able to apply linear algebra tools to varios parctical problems.

#### Course Contents

• Applied Methods of Linear Algebra. An introduction.
• Topic 1. Pseudoinverse matrix, the least squares method, and linear regression
• Topic 2. Polynomial interpolation.
• Topic 3. Metrics and norms.
• Topic 4. Chebyshev polynomials and polynomial approximation.
• Topic 5. Elements of perturbation theory and evaluation of errors.
• Topic 6. Iterative methods and systems of linear equations.
• Topic 7. The eigenvalues problem and the Gershgorin’s theorems.
• Topic 8. Nonnegative matrices and PageRank.
• Topic 9. Functions of matrices.
• Topic 10. Low rank approximation and dimensionality reduction.

#### Assessment Elements

• A homework with a list of individual problems
• Intermediate Test
• Final Exam

#### Interim Assessment

• 2021/2022 2nd module
The final mark is calculated by the formula Total = max(10, (IntermadiateTest + FinalExam)/2 +BonusScore), where IntermadiateTest means the mark for the written test after the 1st module, FinalExam means the mark for the final exam, and BonusScore means the additional score for the students who either prepare a special talk or demonstrate enormous activity during the classes. The talks are evaluated according to the scientific and practical value of the talk, the quality of the presentation, and the number of students participated in the project.

#### Recommended Core Bibliography

• Fuad Aleskerov, Hasan Ersel, & Dmitri Piontkovski. (2011). Linear Algebra for Economists (Vol. 2011). Springer.