Master
2023/2024

# Linear Algebra

Category 'Best Course for Broadening Horizons and Diversity of Knowledge and Skills'

Type:
Compulsory course (Cognitive Sciences and Technologies: From Neuron to Cognition)

Area of studies:
Psychology

Delivered by:
School of Data Analysis and Artificial Intelligence

When:
1 year, 1, 2 module

Mode of studies:
offline

Open to:
students of one campus

Instructors:
Dmitry Frolenkov

Master’s programme:
Cognitive Sciences and Technologies: From Neuron to Cognition

Language:
English

ECTS credits:
6

Contact hours:
40

### Course Syllabus

#### Abstract

The course “Linear Algebra” (in English) covers basic definitions and methods of linear algebra. This course, together with other mathematical courses, provides sufficient condition for students to be ready participate in quantitative and computational modeling at the Master’s program 37.04.01 «Cognitive sciences and technologies: from neuron to cognition».

#### Learning Objectives

- To familiarize students with the subject of mathematics, its foundation and connections to the other branches of knowledge.
- To familiarize students with linear systems and matrices.
- To familiarize students with determinants and volumes.
- To familiarize students with vector spaces and bases.
- To familiarize students with scalar product and norm; distance and angle as derivatives from scalar product.
- To familiarize students with orthogonal and symmetric linear operators.
- To familiarize students with the intersection of linear algebra, calculus, and psychology

#### Expected Learning Outcomes

- Students should be able to check whether vectors are linear independent or not
- Students should be able to construct an orthonormal basis by applying the Gram-Schmidt process.
- Students should be able to construct least square polynomials
- Students should be able to diagonalize a matrix and to evaluate powers of a matrix.
- Students should be able to diagonalize a quadratic form.
- Students should be able to evaluate determinants.
- Students should be able to evaluate the dot product of vectors, to find angles between vectors, to find projection of one vector on another.
- Students should be able to find a transition matrix for change of coordinates
- Students should be able to find basis of vector space
- Students should be able to find coordinates of a vector in a nonstandart basis
- Students should be able to find rank of a matrix
- Students should be able to find the matrix of a linear transformation
- Students should be able to find the new matrix of a lynear transformation after a change of basis
- Students should be able to perform an orthogonall diagonalization of a symmetric operator
- Students should be able to perform basic computations with complex numbers. Students should be able to evaluate powers and roots of a complex number and to evaluate an exponent of a complex number.
- Students should be able to perform fundamental operations with matrices
- Students should be able to solve linear systems using Gauss or Gauss-Jordan method
- Students should be able to solve lynear systems using the theorem of Cramer. Students should be able ti solve homogeneous lynear systems.
- Stugents should be able to find the basis for the kernel and for the range of a linear transformation.

#### Course Contents

- Vectors and matrices
- Solving Linear Systems by Gaussian and Jordan Elimination.
- Determinant definition and properties.
- Rank of a matrix. Inverse matrix
- Complete solution to Ax = b. Linear homogeneous systems
- Eigenvalues and eigenvectors.
- Finite dimensional vector spaces.
- Linear Operators. Matrix Algebra and Matrices of Linear Transformations.
- The kernel and the range of a linear transformation.
- Orthogonality.
- Least-squares polynomials and least square solutions for inconsistent systems.
- Complex numbers.
- Quadratic Forms. Positive definite matrices.
- Matrix Decompositions