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# Linear Algebra

2019/2020
ENG
Instruction in English
3
ECTS credits
Course type:
Compulsory course
When:
1 year, 1, 2 module

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

#### Course Contents

• 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.

#### Assessment Elements

• Test
The retake will consist of the similar tasks and will be evaluated in the same manner.
• Test
The retake will consist of the similar tasks and will be evaluated in the same manner.
• Written Exam
The first retake will consist of the similar tasks and will be evaluated in the same manner. The final mark of the course will also evaluate in the same manner. Sinse there are two exams on the course the second retake will consist of the joint of these two exams. In this case the final mark is equal to the mark obtained during the second retake.
• exam
The first retake will consist of the similar tasks and will be evaluated in the same manner. The final mark of the course will also evaluate in the same manner. Sinse there are two exams on the course the second retake will consist of the joint of these two exams. In this case the final mark is equal to the mark obtained during the second retake.

#### Interim Assessment

• Interim assessment (1 module)
0.3 * Test + 0.7 * Written Exam
• Interim assessment (2 module)
0.7 * exam + 0.3 * Test

#### Recommended Core Bibliography

• Calculus : concepts and methods, Binmore K., Davies J., 2001
• Elementary linear algebra : with supplement applications, Anton H., Rorres C., 2011