General Vector Solution of Fundamental Matrix

Theorem
Let $\Phi \left({t}\right)$ be a fundamental matrix of the system $x' = A \left({t}\right)x$.

Then:


 * $\Phi \left({t}\right) c$ is a general solution of $x' = A \left({t}\right)x$.

Proof
By definition, $\Phi \left({t}\right)$ is non-singular, and therefore has an inverse $\Phi^{-1} \left({t}\right)$.

If $z$ is an arbitrary solution, then $\Phi \left({t}\right) \Phi^{-1} \left({t_0}\right) z( \left({t_0}\right)$ also solves the system and has the same initial condition.

Hence by Existence and Uniqueness Theorem for 1st Order IVPs $\Phi \left({t}\right) \Phi^{-1} \left({t_0}\right) z \left({t_0}\right)$ equals $z$.

Letting $c = \Phi^{-1} \left({t_0}\right) z \left({t_0}\right)$ finishes the proof.