General Vector Solution of Fundamental Matrix

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

Then:


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

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

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

Hence by uniqueness $\Phi(t) \Phi^{-1}(t_0) z(t_0)$ equals $z$.

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