Бакалавриат
2019/2020
Алгебра и геометрия
Лучший по критерию «Новизна полученных знаний»
Статус:
Курс обязательный (Цифровые инновации в управлении предприятием (программа двух дипломов НИУ ВШЭ и Лондонского университета))
Направление:
38.03.05. Бизнес-информатика
Кто читает:
Школа бизнес-информатики
Где читается:
Высшая школа бизнеса
Когда читается:
1-й курс, 1, 2 модуль
Формат изучения:
без онлайн-курса
Преподаватели:
Гончаренко Василий Михайлович
Язык:
английский
Кредиты:
6
Контактные часы:
80
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
Course Contents
- Vector spaces and Homogeneous systemsReal 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 matrixDeterminants 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 numbersComplex 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 EigenvectorsEigenvalues 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 equationsSequences 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 hyperplanesInner 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 sectionsOrthogonal diagonalization of symmetric matrices. Quadratic forms. Quadratic forms and conic sections. Circle, ellipse or hyperbola
- Direct sum and projections. Fitting function to dataThe 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
