We propose a detail study of integral curves or flow lines of a linear vector field in Lorentz [Formula: see text]-space. We construct the matrix [Formula: see text] depending on the causal characters of the vector [Formula: see text] by analyzing the non-zero solutions of the equation [Formula: see text], [Formula: see text] in such a space, where [Formula: see text] is the skew-symmetric matrix corresponding to the linear map [Formula: see text]. Considering the structure of a linear vector field, we obtain the linear first-order system of differential equations. The solutions of this system of equations give rise to integral curves of linear vector fields from which we provide a classification of such curves.