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Магистратура 2016/2017

## Вычислительный практикум в среде Матлаб

Направление: 37.04.01. Психология
Когда читается: 1-й курс, 1, 2 модуль
Прогр. обучения: Когнитивные науки и технологии: от нейрона к познанию
Язык: английский
Кредиты: 3
Материалы по курсу:

### Программа дисциплины

#### Аннотация

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».

#### Цель освоения дисциплины

• 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

#### Содержание учебной дисциплины

• Vectors and matrices
Definition and fundamental operations with vectors. The dot product. Projection. Fundamental opertions with matrices.
• Solving Linear Systems by Gaussian and Jordan Elimination.
The system of linear equations. Gaussian elimination in matrix notation (row reduction). Solving Ax = b by Gaussian and Jordan elimination.
• Determinant definition and properties.
Definition and properties of a determinant of a matrix. Methods for calculating determinant Evaluation of inverse matrix using determinants.
• Rank of a matrix. Inverse matrix
Definition of rank of a mtrix. Definition of inverse matrix. Finding the rank of a matrix and an inverse matrix using Gauss-Jordan method
• Complete solution to Ax = b. Linear homogeneous systems
Complete solution to Ax = b. Solution of linear homogeneous system
• Eigenvalues and eigenvectors.
The characteristic polynomial. Eigenvalues. Eigenvectors. Basis of eigenvectors. Diagonalization of a matrix.
• Finite dimensional vector spaces.
Vector spaces. Span. Linear independence. Basis. Dimension. Coordinatization.
• Linear Operators. Matrix Algebra and Matrices of Linear Transformations.
Linear operator. Difference between matrix and operator. Matrices of linear operator in different bases.
• The kernel and the range of a linear transformation.
Finding the basis for the kernel of a linear transformation. Finding the basis for the range of a linear transformation.
• Orthogonality.
Orthogonal basis. Orthogonalization by the Gram-Schmidt process. Orthogonal matrices. Orthogonal complements. Orthogonal projection onto a subspace. Orthogonal diagonalization.
• Least-squares polynomials and least square solutions for inconsistent systems.
Least-squares polynomials and least square solutions for inconsistent systems.
• Complex numbers.
Complex numbers. Modulus and argument. Complex roots. Elementary functions of complex numbers.
• Quadratic Forms. Positive definite matrices.
Quadratic and forms. The principal axes theorem. Sylvester criteria. Relative extrema of functions of two variables.
• Matrix Decompositions
LU , QR and SVD matrix decompositions.

#### Элементы контроля

• Test
Solving examples similar to the one in the exam.
• Test
• Written Exam
Written exam. Preparation time – 180 min.
• exam

#### Промежуточная аттестация

• Промежуточная аттестация (1 модуль)
0.3 * Test + 0.7 * Written Exam
• Промежуточная аттестация (2 модуль)
0.7 * exam + 0.3 * Test

#### Рекомендуемая основная литература

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

#### Рекомендуемая дополнительная литература

• Linear algebra with applications, Leon S. J., 2002