# LAaG DSBA 2023/2024

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## Teachers and Assistants

Group 231 (M+P+) 232 (П+) 233 234 235
Lecturer Andrey Mazhuga
Teacher Andrey Mazhuga Vladislav Balakirev (tg) Вадим Гизатулин (tg) Vladislav Balakirev (tg) Дарья Башминова (tg)
Consultations Sat, 17:00 -- 21:00, via Zoom
One must notify me beforehand
Assistant Александр Хасанов (tg) Тимофей Мазуренко (tg) Олег Боженов Ксения Астанина Дарья Науменко

## Course Description

The course introduces students to the elements of linear algebra and analytic geometry, provides the foundations for understanding some of the main concepts of modern mathematics. There is a strong emphasis in this course on complete proofs of almost all results.

We will approach the subject from both a practical point of view (learning methods and acquiring computational skills relevant for problem solving) and a theoretical point of view (learning a more abstract and theoretical approach that focuses on achieving a deep understanding of the different abstract concepts).

Topics covered include: matrix algebra, systems of linear equations, permutations, determinants, complex numbers, fields, abstract vector spaces, bilinear and quadratic forms, Euclidean spaces, some elements of analytic geometry, linear operators. It took mathematicians at least two hundred years to comprehend these objects. We plan to accomplish this in one year.

During the academic year, the student will be formally graded on the following:

• two in-class oral tests (O1 and O2);
• two in-class written tests (W1 and W2);
• several quizzes (Q1 and Q2, where Qi is the average grade of all the quizzes in the i-th semester);
• several homework assignments (H1 and H2, where Hi is the average grade of all the homework assignments in the i-th semester);
• two written exams (E1 and E2).

All grades (namely, O1, O2, W1, W2, Q1, Q2, H1, H2, E1, and E2) are real numbers from 0 to 10.

The cumulative course grade for the first semester, C1, is obtained without rounding by the following formula:

C1 = 5/16*O1 + 4/16*W1 + 4/16*Q1 + 3/16*H1.

The intermediate course grade for the first semester, I1, is obtained by the following formula:

I1 = Round1(3/10*E1 + 7/10*C1),

where the function Round1(x) is defined as follows: if the decimal part of x is less than 0.2, the grade is rounded downwards; if the decimal part of x is greater than 0.7, the grade is rounded upwards; if the decimal part of x is from the interval [0.2;0.7] and the student's seminar attendance during the first semester is not below 66%, the grade is rounded upwards; otherwise the grade is rounded downwards.

The cumulative course grade for the second semester, C2, is obtained without rounding by the following formula:

C2 = 5/16*O2 + 4/16*W2 + 4/16*Q2 + 3/16*H2.

The intermediate course grade for the second semester, I2, is obtained by the following formula:

I2 = Round2(3/10*E2 + 7/10*C2),

where the function Round2(x) is defined as Round1(x) but with "during the first semester" replaced by "during the second semester".

The final grade for the course, F, is obtained by the following formula:

F = Round(1/4*I1 + 3/4*I2),

where the function Round(x) is defined as Round1(x) but with "during the first semester" replaced by "during the academic year".

The final grade for the course is included in a diploma supplement.

## Lecture Notes

Module 2

• Lecture 13 (06.12.2023) Vector spaces (Part II); Main theorem on linear dependence; a basis of a vector space; dimension of a vector space; finite and infinite dimensional vector spaces; some main properties of finite dimensional vector spaces.
• Lecture 12 (29.11.2023) Vector spaces (Part I); a subspace of a vector space; linear combinations; the linear span; trivial and non-trivial linear combinations; linearly dependent and linearly independent sets.
• Lecture 11 (22.11.2023) Polynomials; the degree of a polynomial; division with remainder; a root of a polynomial; factor theorem; the multiplicity of a root; algebraically closed fields; fundamental theorem of algebra; interpolation polynomials in the Lagrange form.
• Lecture 10 (15.11.2023) Complex numbers (part II); main properties of the conjugate and the absolute value of a complex number; polar (= trigonometric) form of a complex number; de Moivre's Formula; the set of n-th roots of a complex number, its description.
• Lecture 9 (11.11.2023) The determinant of a Vandermonde matrix; fields; the field of complex munbers; Cartesian and algebraic forms of a complex number; the absolute value (=module) and the argument of a complex number, the complex conjugate.
• Lecture 8 (01.11.2023) Block matrices; the determinant of a block matrix; minors and cofactors of a matrix; Laplace expansion; false expansion; the adjugate of a matrix; Cramer's rule.

Module 1

• Lecture 7 (18.10.2023) Determinant of an elementary matrix; determinant of a product of matrices; determinant test for invertibility.
• Lecture 6 (11.10.2023) Determinant; the Leibniz Formula; Sarrus' Rule; determinant of matrix transpose; three main properties of the determinant; the determinant of a matrix with a zero row or column.
• Lecture 5 (04.10.2023) Permutations; two-line notation of a permutation; the sign of a permutation; even and odd permutation; transpositions.
• Lecture 4 (27.09.2023) Systems of linear equations (SoLE); homogeneous, inhomogeneous, consistent, and inconsistent SoLE; the matrix form of a SoLE; leading and free variables; the augmented matrix of a SoLE; a general algorithm for solving SoLE.
• Lecture 3 (20.09.2023) Elementary row matrix operations; elementary matrices; elementary row operations as matrix pre-multiplication; reduced row echelon form; Gaussian elimination.
• Lecture 2 (13.09.2023) Matrix transposition; symmetric and skew-symetric matrices; inverse of a matrix; invertible (=non-singular) matrices; the trace of a matrix; main properties of the matrix transposition; the trace.
• Lecture 1 (06.09.2023) Matrices, main definitions; special matrices (square matrices, triangular matrices, identity matrices, zero matrices); matrix scalar multiplication; matrix addition; matrix multiplication; main properties of these operations.

## Homework

### The homework for group 231:

Module 2

• HW 12 (release: 03.12.23; deadline: 10.12.23) (seminar notes)
• HW 11 (release: 25.11.23; deadline: 03.12.23)
• HW 10 (release: 17.11.23; deadline: 26.11.23) (seminar notes)
• HW 9 (release: 10.11.23; deadline: 19.11.23)
• HW 8 (release: 03.11.23; deadline: 12.11.23)

Module 1

Module 2

Module 1

### The homework for group 233:

Module 1

• [??? HW 1] (release: ???; deadline: ???) ([??? seminar notes])

### The homework for group 234:

Module 2

Module 1

• [??? HW 1] (release: ???; deadline: ???) ([??? seminar notes])

### The homework for group 235:

Module 1

• [??? HW 1] (release: ???; deadline: ???) ([??? seminar notes])

## Results

231 232 233 234 235