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# Algebra and Geometry

2019/2020
ENG
Instruction in English
6
ECTS credits
Course type:
Compulsory course
When:
1 year, 1, 2 module

### Course Syllabus

#### Abstract

In the process of studying the discipline, students will become familiar with theoretical foundations and basic methods of solving tasks on the following topics • Systems of linear equations. Row operations and Gaussian elimination. Vectors and Matrices. Linear spaces. Homogeneous systems and null space. • Matrix inversion and determinants. Leontief input-output analysis. • Complex numbers and their properties. • Eigenvalues and eigenvectors. Diagonalization of matrices. • Sequences, series and difference equations. Coupled first-order difference equations. Their applications in economics and finance • Inner product and orthogonality. Lines in R2, planes and lines in R3, lines and hyper-planes in Rn . • Orthogonal diagonalisation. Quadratic forms and conic sections • Direct sum and projections. Fitting function to data: least squares approximation

#### Learning Objectives

• Provide students with an understanding of key concepts and methods of algebra and geometry for under-standing other practical courses, related to data analysis and programming

#### Expected Learning Outcomes

• Upon completion of the course, students should be aware of the basics of linear algebra and geometry; key concepts and approaches to their study. The will be able to formalize the problem from subject area, choose the adequate methods of solutions, perform essential calculations and to interpret the results. Also they learn how to be able to formalize the problem from subject area, choose the proper methods of solutions, find and to interpret the results obtained.

#### Course Contents

• Theme 1. System of linear equations and matrices.
Introduction to Systems of Linear Equations. Gaussian Elimination. Matrices and Matrix Operations. Algebraic Properties of Matrices. Powers of the matrix. Transpose matrix. Inverse matrix. Method for finding the inverse matrix with row operations. Symmetric matrices. Vectors in palne and space. Inner product.
• Vector spaces and Homogeneous systems
Real Vector Spaces and Subspaces. Linear Independence and Dependence of vectors. Coordi-nates and Basis. Dimension. Solution Spaces of Homogeneous Systems. Change of Basis. Row Space, Column Space, and Null Space. Rank, Nullity and the Fundamental Matrix Spaces
• Determinants and inverse matrix
Determinants of matrices. Finding determinants by Cofactor Expansion. Evaluating Determi-nants by Row Reduction. Properties of Determinants. Cramer’s Rule. Nondegenerate matrix and ex-istence of inverse. Adjoint matrix. Using adjoint matrix to find inverse matrix. Leontief input-output analysis
• Complex numbers
Complex numbers. Complex conjugate. Algebra of complex numbers. The complex plane. The polar form of a complex number. The modulus and the argument of a complex numbers. Complex vector spaces and complex matrices
• Eigenvalues and Eigenvectors
Eigenvalues and Eigenvectors. Diagonalization of a square matrix. Eigenvalues and Eigenvectors of Matrix Powers. Determinants and eigenvalues. Similar matrices. Finding the power of a matrix using diagonalization
• Difference equations
Sequences and progressions. Compound interest. Frequent compounding Series and financial applications. First-order difference equations and their solution. Long-term behavior. The cobweb model. Second-order difference equations. Behavior of solutions. Economic applications
• Euclidean vector spaces. Lines, planes and hyperplanes
Inner product and orthogonality. Euclidean vector spaces. Lines in R2, planes and lines in R3, lines and hyperplanes in Rn. Geometry of linear systems
• Quadratic forms and conic sections
Orthogonal diagonalization of symmetric matrices. Quadratic forms. Quadratic forms and conic sections. Circle, ellipse or hyperbola
• Direct sum and projections. Fitting function to data
The direct sum of two subspaces. The orthogonal complement of a subspace. Orthogonal com-plements of null spaces and ranges. Projections. Orthogonal projections. Orthogonal projection onto the range of a matrix. Minimizing the distance to a subspace. Fitting functions to data: least squares ap-proximation

#### Assessment Elements

• Control work 1
• Control work 2
• Homeworks
• Final exam

#### Interim Assessment

• Interim assessment (2 module)
0.18 * Control work 1 + 0.18 * Control work 2 + 0.36 * Final exam + 0.28 * Homeworks

#### Recommended Core Bibliography

• Elementary linear algebra : with supplement applications, Anton H., Rorres C., 2011