Take the tour ×  There is a remark which gets used many times in
a theorem I am trying to work through:  If $X(t)$ is a
fundamental matrix of $x'(t) = A(t)x(t)$ (1), then $x(t) =
X(t)X^{-1}(t_{0})x_{0}$.  In this setting, we are taking $x_{0}
= x(t_{0})$.  To be honest, I'm not exactly sure what it is
claiming exactly. Is just just a closed form of a particular
solution to (1)? or is it a general solution to (1)? Since
fundamental matrices can differ by multiples of constant
matrices, this couldn't be describing all the solutions, right?
share|improve this question From what book is this? –  Ｊ. Ｍ. Oct
30 '11 at 0:47 We aren't using a book. It's just from my course
notes. I'm referring to the first step in the proof of (a) of
Theorem 3.1 on page 4.
math.ualberta.ca/~mli/courses/current/524/chapter3.pdf –  Kyle
Oct 30 '11 at 1:34 add comment  1 Answer  up vote 5 down vote
accepted  The columns of a fundamental matrix span the space of
solutions. In other words, if the columns of $X(t)$ are
$x_1(t),\dots,x_n(t)