Introduction to basic algebraic structures. Fields, rings, examples.

Vector spaces. Linear combinations, linear span, linear independence, generating sets, finitely and infinitely generated fields, basis, coordinates (with respect to a basis), dimension, theorem on the dimension of the join and meet; examples.

Homomorphisms of vector spaces. Basic properties of homomorphisms, special types of homomorphisms, the theorem on the dimension of the kernel and the image; examples.

Homomorphisms and matrices. The matrix of a homomorphism, compositions of homomorphisms and product of matrices, transformation of coordinates of a vector, rank of a matrix, elementary transformations, methods for calculating the rank of matrix, transformations of matrices, inverse matrix; examples.

Systems of linear equations. Solvability, the space of solutions and its dimension, the theorem of Frobenius, Gauss elimination method; problems; examples.

An introductory course in linear algebra (introduction to basic algebraic structures, vector spaces, homomorphisms, homomorphisms and matrices, systems of linear equations).