Fundamental matrix linear system equivalent to linear matrix system

[member 731016]

Homework Statement

In my lecture notes I have $Φ'(t) = AΦ(t) ⟷ x'(t) = Ax$. I am trying to understand why.

Relevant Equations

$Φ'(t) = AΦ(t) ⟷ x'(t) = Ax$

My working is ,
Consider case where the there are two linearly independent solutions
$x'(t) = c_1x' + c_2y' = A(c_1x + c_2y)$
$(x'~y')(c_1~c_2)^T = A(x~y)(c_1~c_2)^T$

Then cancelling coefficient matrix I get,
$(x'~y')= A(x~y)$
$Φ'(t) = AΦ(t)$ by definition of 2 x 2 fundamental matrix

Does someone please know whether this proof is correct?
Thanks!

 

[fresh_42]

What is the difference between $\phi(t)$ and $x(t)$? How are both defined?

 

[pasmith]

What do you get if you differentiate $x(t) = \Phi(t)x_0$ for constant $x_0$?