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.