# Lecture course in Higher Algebra

Faculty of Applied Mathematics & Control Processes, St.Petersburg State University

Lecturer: Prof. Alexei Uteshev

# Part I. Solution of Equations and Systems of Equations

## Introduction to Number Theory

Binomial theorem. Pascal's triangle.

The greatest common divisor (GCD). Euclidean algorithm. Linear representation for GCD, the continuant.

Divisibility. Coprime numbers. Prime numbers. Factorization.

Euler's totient function.

Congruence. Modular arithmetic. Square-and-multiply algorithm. Theorems by Fermat and Euler.

Linear congruences solving. Modular inverse. Wilson's theorem. Chinese remainder theorem.

## Complex numbers

Basic notions.

De Moivre's formula. Trigonometric equalities.

Root extraction. Roots of unity. Primitive roots.

## Polynomial and rational functions

Basic notions, Horner's process.

The fundamental theorem of Algebra (formulation). Viète's formulas. Factorization. Multiplicity of the root.

Solving by radicals: cubic and quartic equations. Cardano's formula for the cubic equation with real coefficients: casus irreducibilis.

Divisibility. The greatest common divisor. Euclidean algorithm. Relatively prime polynomials, Bézout's identity.

Taylor expansion.

Multiple root finding.

Polynomials with real coefficients. Bounds for the roots: by Maclaurin, Lagrange and Newton. Descartes rule of signs.

Numerical methods for solving algebraic equations: the Ruffini-Horner method, the Lagrange (continued fractions) method, Newton's iterations.

Rational functions.

Multivariate polynomials, their representations. Taylor's expansion, extrema conditions. Algebraic equations.

## Systems of linear equations, matrices and determinants

Systems of linear equations. Gaussian elimination.

Matrices and their elementary properties.

Definition of the determinant. Permutations.

Fundamental properties of the determinant.

Minors and cofactors. Expansion of the determinant.

Cramer's formulas.

Laplace's theorem

The Binet-Cauchy theorem. Cauchy's inequality.

Special types of determinants ( Vandermonde, Hankel, with integer entries, characteristic polynomial)

Matrix inverse.

Rank.

Conditions for the consistency of linear system. The Kronecker-Capelli theorem. Complete solution. Homogeneous linear equations. The fundamental system of solutions.

## Interpolation

Polynomial interpolation: methods by Lagrange and Newton.

The least squares method.

Overdetermined system of equations. Pseudoinverse.

## Elimination Theory

Resultant and its linear representation.

Tschirnhaus transformation.

Elimination procedure for the system of bivariate equations. Bézout's theorem (on the number of solutions in $\mathbb C^2$).

The fundamental theorem of algebra (proof).

Basic notions. Canonical form. Lagrange's method

Sylvester's law of inertia. Equivalence of quadratic forms.

Sign-definiteness of quadratic forms. Sylvester's criterion.

## Localization of Roots of a Polynomial

Sturm's sequence.

Hankel matrices in the problem of root localization: Newton's sums, theorems by Jacobi and Joachimsthal.

Roots in special domains of the complex plane: criteria for the stability and discrete stability of a polynomial.

## Abstract Algebraic Structures

Binary operation. Group, subgroup. Generators. Lagrange's theorem.

Multiplication (Cayley's) table. Factor group. Isomorphism.

Rings, fields, algebras. Quaternions.

# Part II. Linear Spaces and Mappings

## Linear Spaces and Manifolds

Basic notions. Isomorphism.

Linear dependency, basis. Relative linear dependency. System of linear homogeneous and its fundamental system of solutions

Sum and intersection of subspaces. System of linear inhomogeneous equations.

Direct sum of subspaces

Change of basis matrix

## Inner Product Spaces

Basic notions.

Orthogonal basis. The Gram-Schmidt orthogonalization process.

Gram's determinant. Nonnegativity, distance evaluation, Hadamard's inequality for the determinants, volumes

## Linear Mappings

Kernel and image

Matrix of a linear mapping

Operator

Invariant subspaces. Eigenvectors and eigenvalues of an operator. Characteristic polynomial. Non-derogatory matrices.

Properties of a characteristic polynomial. The Cayley-Hamilton theorem.

Symmetric matrix and properties of its eigenvalues and eigenvectors.

Generalized eigenvectors. Jordan canonical form (JCF).

## Applications of the Jordan canonical form

Matrix polynomial

Linear difference equation

Probability theory. Gambler's ruin problem. Markov chain.

Analytic functions of a matrix. Matrix norm. Linear ordinary differential equations. Notion of stability

## Numerical methods of linear algebra

Characteristic polynomial evaluation. Leverrier's method. Krylov's method.

Partial eigenvalue problem

# References

1. Faddeev D.K., Faddeeva V.N. Computational Methods of Linear Algebra. Freeman. 1963

2. Grossman I., Magnus V. Groups and Their Graphs. MAA. 1964

3. Horn R.A.,Johnson C.R. Matrix analysis. Cambridge University Press. 1986

4. Strang G. Linear Algebra and its Applications. Cengage learning. 2006

5. Uspensky J.V. Theory of Equations. McGraw-Hill. 1948

6. Wilkinson J.H. The Algebraic Eigenvalue Problem. Clarendon Press. Oxford. 1965

algebra2/course/prog_eng.txt · Последние изменения: 2023/07/01 15:18 — au