On linear representations of some extensions

Yu. I. Merzljakov developed a method of splittable coordinates which helps to verify the linearity of some groups, he established some fundamental results using this method.
In this paper we use the method of splittable coordinates and find some sufficient condition under which the semi–direct product of two linear groups is linear. As consequence we get linearity of some HNN-extensions of a free group, linearity of the holomorph of the braid group $B_n$, $n\geqslant 2$, and free group $F_2$ and linearity of some Artin groups. In all cases we construct faithful linear representations in the explicit form.