Task from an exam:
Given is the real vector space where $a \in \mathbb{R}$ is fixed:
$V = \left\{ f: \mathbb{R} \rightarrow \mathbb{R}: f(x) = \lambda_{1}e^{x}\sin(ax)+ \lambda_{2}e^{x}\cos(ax) \text{ for } x \in \mathbb{R}, \lambda_{1}, \lambda_{2} \in \mathbb{R} \right\}$
with the basis $B=(f_{1},f_{2})$, whereby $\text{ }$ $\text{ }$$f_{1}(x)=e^{x}\sin(ax), \text{ } \text{ }f_{2}(x)= e^{x}\cos(ax)$.
And because every element of this vector space is characterized by an ordered pair $(\lambda_{1}, \lambda_{2}) \in \mathbb{R}^{2}$ of coefficients, with allocation $f=\lambda_{1}f_{1}+\lambda_{2}f_{2} \mapsto (\lambda_{1}, \lambda_{2})$, every linear mapping of $V$ in $V$ can be seen as a $2\times 2$ matrix.
Let $L: V \rightarrow V$ be the linear mapping which assigns every function $f \in V$ its derivative $f' \in V$ to $x$.
State this re-presentation $G_{BB}(L) \in \mathbb{R}^{2 \times 2}$ of $L$ in the basis $B$.
I have troubles solving this task especially because I don't understand the notation $G_{BB}(L)$. So I think $L$ stands for first derivative and then we take it for $G_{BB}$ which is the representation? :s