Let $T:V_3\to V_3 $ be a linear transformation such that
(both bases are usual unit coordinate vectors)
Now choose both bases to be $(e_1,e_2,e_3)$ where $e_1=(2,3,5)$, $e_2=(1,0,0)$, $e_3=(0,1,-1)$
Determine the matrix of T reative to the new bases.
My book says, that to determine the matrix of T relative to choice of bases, I need to transform each basis element of the first space and express it as a linear combination of the basis element of the second space. First of all, I express the transformation as follows:
Then for example $T(e_1)=-e_1+e_2-e_3=(-1,-4,-4)$ (right?)
Then I should express it a combination of $e_1,e_2,e_3$. If I do that, if I understand it correctly, I shuould get the first column of the matrix. However, the first column of the matrix has corresponding vaues of $2,1,2$ (I know this from the answers) and obviously something is wrong with my solution as these numbers does not correspond to the combination I am supposed to find.
I missunderstood something and looking forward to hints of how should I find the matrix to the new chosen bases. Thank you!