Find the matrix for d/dx acting on the vector space 
Im not sure if Im going about this problem the right way and any help would be appreciated.  
I took the derivative of the vector space and ended up with  $c_1e^{2x}+2c_1e^{2x}+ 2c_2e^{2x}$.


If the original ordered basis only has two components does that mean that the final matrix only has two rows? Will my matrix be in $\Bbb{R}^2$?



Thanks

Make your linear map act on the given basis and write the outcome as a linear combination of the same basis:
$$\begin{align*}&\frac d{dx}(xe^{2x})=e^{2x}+2xe^{2x}=&\color{red}2\cdot xe^{2x}+\color{red}1\cdot e^{2x}\\{}\\
&\frac d{dx}(e^{2x})=2e^{2x}=&\color{red}0\cdot xe^{2x}+\color{red}2\cdot e^{2x}\end{align*}$$
Thus the matrix for the linear operator $\;\frac d{dx}\;$ with respect to the given basis $\;B\;$ is
$$A:=\begin{pmatrix}2&0\\1&2\end{pmatrix}$$
