Modern verification commonly models software with Boolean logic and a system of linear inequalities over reals and over-approximates the reachable states of the model with Craig interpolation to obtain, for example, candidates for inductive invariants. Interpolants for the linear system can be efficiently constructed from a Simplex refutation by applying the Farkas’ lemma. However, this approach results in the worst case in incompleteness, since Farkas interpolants do not always suit the verification task. This work introduces the decomposed interpolants, a fundamental extension of the Farkas interpolants obtained by identifying and separating independent components from the interpolant structure using methods from linear algebra. We integrate our approach to the model checker Sally and show experimentally that a portfolio of decomposed interpolants results in immediate convergence on instances where state-of-the-art approaches diverge. Being based on the efficient Simplex method, the approach is very competitive also outside these diverging cases.