Normally, if we have an $n$-dimensional linear system of differential equations, we write $$ \dot x=Ax, $$ with $x\in\mathbb R^n$. Now I am wondering why the convention isn't to write $$ x=A\dot x. $$ It seems to me we want an experssion of $\dot x_i$ for each $i$ in terms of the $x_i$'s - but why not the other way around?
The reason I'm asking this is because when we have an $n$th degree differential equation, they teach us to write it in a system of the form $\dot x=Ax$. I didn't see why it had to be the case that it has this form.