A fundamental problem in one of the most popular models for continuous environments; i.e., linear dynamical systems, is to learn dynamics matrices from unstable state trajectories. While this problem is well-studied for a single system, little is known about multitask learning methods that utilize potential commonalities to estimate the dynamics matrices more accurately. The longstanding obstacle is that idiosyncratic instabilities nullify the benefit of sharing data across systems. We address this issue by introducing the new method of \emph{big random control actions}, and develop a novel performance analysis for that. To the authors' knowledge, this is the first theoretical guarantee for multitask learning under instability when system matrices are \emph{unknown} linear combinations of \emph{unknown} bases matrices. The techniques can be extended to multitask learning problems in other settings with non-stationary or temporally dependent data.