Fundamental solution of linear system of ODEs I struggle to understand what the fundamental solution is supposed to be.
Specifically it's about a linear system of homogen ODEs with constant coefficents of the form: $\dot{\textbf{F}}=\textbf{AF}$ where $\textbf{F},\dot{\textbf{F}}:\mathbb{R} \to \mathbb{R^n}, \textbf{A} \in \mathbb{R} ^{n \times n}$.
In the lecture we were told $\Phi(t) = Exp(\textbf{A}t) : \mathbb{R} \to  \mathbb{R} ^{n \times n} $ is the fundamental solution of the system, since symbolically $\Phi'(t)=\textbf{A}\Phi(t)$. With the definition I see this makes sense but why is the solution a matrix instead of a vector? I really don't understand the relatonship between the fundamental solution and $\textbf{F}$ nor what the fundamental solution even is? (Calculus in Engineering is sometimes a bit short with explanations...)
 A: This is a subtle point of terminology.  The solution to the first-order, linear, homogeneous system $\dot{f}(t) = A\,f(t)$ where $f,\dot{f} : \mathbb{R} \rightarrow\mathbb{R}^{n\times1}$ and $A\in\mathbb{R}^{n\times n}$ has the solution
$$
f(t) = \sum_{k=1}^{n} c_k f_k(t)
$$
where $c_k \in \mathbb{R}$ and $f_k: \mathbb{R} \rightarrow\mathbb{R}^{n\times1}$   $\forall k$.  In others words, the solution to the above linear system of dimension $n$ is a linear combination of $n$ independent solutions (ideally).  We can re-write the summation in linear algebra notation as
$$
f(t) = \Phi(t)\,c
$$
where 
$$
\Phi(t) = [f_1,f_2,\ldots,f_n]\quad\mbox{and}\quad c=[c_1,c_2,\ldots,c_n]^{\intercal}.
$$
Therefore, just as $y = \exp(x)$ may be called a fundamental solution to $\dot{y} = y$ as it encompasses all of the solution's behavior up to a multiplicative scalar determined by initial conditions, the matrix of solutions $\Phi(t)$ is sometimes called the fundamental solution to the linear system $\dot{f}(t) = A\,f(t)$ as it encompasses all of the solutions' behaviors up to a multiplicative vector determined by initial conditions.
