ABSTRACT

This text contains more than 2000 exercises in algebra. These exercises are currently used in teaching a fundamental course in algebra in the Department of Mechanics and Mathematics at Moscow State University. The text is divided into three parts, which correspond to three semesters of study. Each section contains not only standard exercises, but also more difficult exercises at the end of some sections, these more challenging exercises being marked with asterisks. At the end of the book, results of calculations, a list of notations and basic definitions are given.

part 1|98 pages

Foundations of Algebra

chapter 1|12 pages

Sets and maps

chapter 2|16 pages

Arithmetic spaces and linear equations

chapter 3|17 pages

Determinants

chapter 4|11 pages

Matrices

chapter 5|15 pages

Complex numbers

chapter 6|22 pages

Polynomials

part 2|106 pages

Linear Algebra and Geometry

chapter 7|13 pages

Vector spaces

chapter 8|16 pages

Bilinear and quadratic functions

chapter 9|20 pages

Linear operators

chapter 10|20 pages

Metric vector spaces

chapter 11|6 pages

Tensors

chapter 12|28 pages

Affine, Euclidean, and projective geometry

part 3|107 pages

Basic Algebraic Structures

chapter 13|40 pages

Groups

chapter 14|41 pages

Rings

chapter 15|23 pages

Elements of representation theory