In this thesis we mainly deal with a \(2\times 2\) linear systems of ordinary differential equations. In the first chapter, we revise the basic facts about the matrix representation of the linear operator and about the Jordan basis, with the aim of finding a linear change of variables in which the system has a simpler (possibly diagonal) form and is easier to solve. In the second chapter, we give a geometric interpretation of \(2\times 2\) linear systems of ordinary differential equations using the field of directions. We introduce basic terms such as an integral curve and the phase portrait (the trajectories of the system displayed in \(\mathbb{R}^2\)). As a preparation for the statement of the fundamental theorem about the explicit solution of the linear system, in the third chapter we introduce the concept of the matrix norm, which gives the space of the square matrices the structure of the normed space and we are able to study the convergence of the sequences and series of matrices. Then we express a fundamental theorem about the solution of the linear system in which the solution is expressed as a matrix exponential in terms of the convergent series in the space of matrices. Depending on the form of the Jordan matrix, i.e. on the type of its eigenvalues, we get different types of phase portraits. At the end of the chapter we classify linear systems according to the types of eigenvalues. The explicit computations of solutions are explained from the viewpoint of qualitative analysis, i.e. by analyzing the trajectories of the system using fields of directions and the geometric interpretation of the system. In the fourth chapter, we define the linear approximation of the nonlinear system. We state the Hartman-Grobman’s theorem which under some circumstances states the similarity of the phase portrait of the system and of its linear approximation. Thus we give an application of the linear theory to non-linear systems, and illustrate on an example.