Fundamental matrix linear system equivalent to linear matrix system

[ChiralSuperfields]

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 [itex]x(t) = \Phi(t)x_0[/itex] for constant [itex]x_0[/itex]?