Sorry I might have kind of a dumb question here.
Let $ f:\mathbb{R} \rightarrow \mathbb{R}$ and consider a linear differential equations of the form $$ a_{n}f^{(n)} + a_{n-1} f^{(n-1)} + ... + a_{2} f^{(2)} + a_1 f^{(1)}+a_0f= 0$$ one could think of the space of solutions as the kernal of the linear operator $$a_{n}D^n + a_{n-1}D^{n-1}+...+a_{2}D^2 + a_{1}D + a_{0} $$ which acts on the space of all $ n $-th order differentiable functions. However, when thinking about a system of linear differential equations of the form $$ \dot{f} = Af $$ where $ f $ is an $ n $-th dimensional vector $ \Big( f_{1}(t), .. f_{n}(t) \Big) $ and $ A $ is an $ n \times n $ matrix I get kind of confused. So the question is simple: does $A$ in this case corresponds to any linear operator $T$ acting on any space that is not $\mathbb{R}^{n}$? Or is it right for me to think of $A$ geometrically as an endomorphism on $\mathbb{R}^{n}$ in its own right. Cause what happen is that I am aware that every linear operator defined on arbitrary vector space over $\mathbb{K}$ can be represented isomorphically as a matrix defined on the coordinates. But in this case think it doesn't seems likely but the then the theory I stated in the very beginning gives me some doubt. So basically I am asking if this matrix $A$ has anything to do with the general theory in terms of the differential operator $D$ or am I just too blind to see it.
Thanks in advance!