Content: Let $\phi^j(t)$ be solution of initial value problem $x'=Ax$, where $x,x'$ are n-dimensional vectors. Also we suppose that $x(0)=e^j$, where $e^j$ is vector with $n-1$ zeros, and 1 on j-th place. Task is to show that: $$e^{At}=(\phi_1(t),...,\phi_n(t))$$
Note: I've got totaly stuck with this exercise, while going through Differential Equations and Their Applications, by M. Braun. I'm not especially keen on linear algebra, and maybe thats the reason why I've got no clue where to start, or what to do with this proof, despite trying to go by definitions. But there I don't see why series on evry dimension of $e^{At}$ have to converge to $(\phi_1(t),...,\phi_n(t))$. I will greatly appriciate any hint, help or even example of solution of this, or similar problem. I'm undergraduate at 2nd semester of my 2nd year of Math studies, so please do not throw anything explicitly harsh to understand. Also English is not my main language, so sorry for any inconviniences in communication, and mistakes made in it. Thanks in advance!