General Vector Solution of Fundamental Matrix

Theorem

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

Then:

$\map \Phi t c$ is a general solution of $x' = \map A t x$.

Proof

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

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

Hence by Existence and Uniqueness Theorem for 1st Order IVPs $\map \Phi t \, \map {\Phi^{-1} } {t_0} \, \map z {t_0}$ equals $z$.

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

$\blacksquare$