معلومات المقرر

    Algebra is a field of mathematics that enables us to express the properties of operations and the treatment of equations and results in the study of algebraic structures. Depending on the level of study considered, it can be described as general algebra (abstract) and linear algebra. Abstract algebra is concerned with:

    ⋅ Generalized arithmetic, extending the usual operations on numbers to different objects or quantities,

    ⋅ Equation and polynomial theory,

    ⋅ Algebraic structures.

    Algebraic structures have historically appeared in several fields of mathematics, and have not been studied separately. This is why general algebra has many connections with all branches of mathematics, a large number of types of algebraic structures verify different axioms (groups, rings, fields, vector spaces,...etc.). For these different types of structures, we define a concept of homomorphism and constructions of analogous structures or structures with similar properties (substructures, quotients, products, etc.).

    For the second part, historically linear algebra began with the study of linear systems. Approached as early as 1678 by Leibnitz, Maclaurin in 1748 gave formulas for solving two or three unknowns, completed in the general case by Cramer in 1754. From there, Vandermonde and then Laplace came up with the idea of defining a determinant of order n by recurrence on n∈ℕ, developing it with respect to a row or column. In addition, in research arithmétiques, Gauss adopted a tabular notation to designate a linear transformation: matrix notation. This passage was to suggest to Cauchy the rule for the product of two determinants, published in 1815.

    Given the interest in this vast field of mathematics, we propose this document be considered as a series of solved exercises distinguished to students of 1st year Mathematics Informatics, this document is composed of nine chapters, the first begins with logic and sets, which are fundamental in mathematics, then binary relations defined on a set are presented, followed by the study of algebraic structures and the polynomial ring, In the linear algebra section, we start with vector spaces, then linear maps, matrices and finally determinants and systems of linear equations.

    This document is organized as follows: in each chapter, a general reminder of the notions required in the chapter is given, followed by a series of solved exercises.