Global solution to the ODE if $f$ is linear

Question

I was reading an ODE textbook, and there was a part where it was explained under which condition IVP:

$$\dot{x}=f(x); \; x(t_0)=x_0$$

with $f:\mathbb{R^n}\rightarrow\mathbb{R^n}$ has local, global and unique solutions, most of the things were rather clear but there was one question regarding the effect of the linearity where I got confused. So the question is:

If we assume that $f$ linear, is there for all $t\in\mathbb{R}; \; x_0\in\mathbb{R^n}$ a global solution to the IVP?

I couldn´t find any theorem that would imply it but I might be missing something.

Answer 1

The linear IVP $$\dot{x} = A x;\ x(t_0) = x_0$$ has an explicit global solution $$x = \exp((t-t_0) A)\; x_0 $$ where $\exp$ is the matrix exponential.