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Linear Algebra

2020/2021
Учебный год
ENG
Обучение ведется на английском языке
4
Кредиты

Преподаватель

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. • Complex numbers and their properties. • Eigenvalues and eigenvectors. Diagonalization of matrices. • 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
Learning Objectives

Learning Objectives

  • Provide students with an understanding of key concepts and methods of linear algebra for understanding other practical courses, related to data analysis and programming
Course Contents

Course Contents

  • 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
  • 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
Assessment Elements

Assessment Elements

  • non-blocking Control work 1
  • non-blocking Control work 2
  • non-blocking Homeworks
  • non-blocking Final exam
    The exam is ONLINE in written form. It is taken on Ms Teams platform. Students are required to join a session 15 minutes before the beginning. The computers must meet the following technical requirements: https://docs.microsoft.com/ru-ru/microsoftteams/hardware-requirements-for-the-teams-app A student is supposed to follow the requirements below: check your computer for compliance with technical requirements no later than 5 days before the exam; sign in with your corporate account (@edu.hse.ru); check your microphone, speakers or headphones, webcam, Internet connection (we recommend connecting your computer to the network with a cable, if possible); prepare the necessary writing equipment, such as pens, pencils, pieces of paper, and others. Disable applications on the computer's task other than the MS Teams application or the browser that will be used to log in to the MS Teams platform. Students are not allowed to: turn off the video camera; use notes, textbooks, and other educational materials; leave the place where the exam task is taken (go beyond the camera's viewing angle); look away from your computer screen or desktop; use smart gadgets (smartphone, tablet, etc.) involve outsiders for help during the exam, talk to outsiders during the examination tasks. Students are allowed to: write on a piece of paper, use a pen for making notes and calculations; use a calculator (not on mobile); turn on the microphone to answer the teacher’s questions; ask a teacher for additional information related to understanding the exam task. A short-term communication failure during the exam is considered to be the loss of a student's network connection with the MS Teams platform for no longer than 1 minute. A student cannot continue to participate in the exam, if there is a long-term communication failure appeared. The retake procedure is similar to the exam procedure. In case of long-term communication failure in the MS Teams platform during the examination task, the student must notify the teacher, record the fact of loss of connection with the platform (screenshot, a response from the Internet provider). Then contact the manager of a program with an explanatory note about the incident to decide on retaking the exam.
Interim Assessment

Interim Assessment

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

Bibliography

Recommended Core Bibliography

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

Recommended Additional Bibliography

  • Mathematics for economics and finance : methods and modelling, Anthony, M., Biggs, N., 2012
  • Mathematics for economists, Simon, C. P., Blume, L., 1994