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

First apply the $\frac{d}{dx}$operator on the basis to obtain the transformed basis:
$$D 
  \left[ {\begin{array}{c}
   1  \\       0  \\      \end{array} } \right] = \left[ {\begin{array}{c}
   2  \\       1  \\      \end{array} } \right]$$
$$D 
  \left[ {\begin{array}{c}
   0  \\       1 \\      \end{array} } \right] = \left[ {\begin{array}{c}
   0  \\       2  \\      \end{array} } \right]$$
Hence, the transformation of the Identity matrix, and hence the matrix representation of the transformation is given by
$$D 
  \left[ {\begin{array}{cc}
   1 & 0  \\       0 & 1  \\      \end{array} } \right] = DI = D = \left[ {\begin{array}{c}
   2 & 0  \\       1 & 2  \\      \end{array} } \right]$$
