For a linear transformation F we're given that F(0,1,0)=(1,1,0), F(0,0,1)=(1,-1,2), and that (1,1,1) is an eigenvector with eigenvalue=1. Find the transformation matrix for F.
I know this can be solved by just using linearity: (1,0,0) = (1,1,1)-(0,1,0)-(0,0,1) so using linearity of the transformation, F((1,0,0)) = F((1,1,1)-(0,1,0)-(0,0,1)) gives the image of the "missing" basis vector and thus the matrix (call it $A$), which is: $A$= \begin{pmatrix}-1&1&1\\ 1&1&-1\\ -1&0&2\end{pmatrix}
However, this also seemed straightforward to solve by changing to a new basis given by: e_1= (0, 1, 0), e_2 = (0, 0, 1) and e_3 = (1, 1, 1). I'm thinking that in the new basis, the transformation matrix ($\hat{A}$) must be the following, given the information about the images of the vectors under the transformation given: $\hat{A}$=\begin{pmatrix}1&1&1\\ 1&-1&1\\ 0&2&1\end{pmatrix}
If we call the coordinate transformation matrix S=\begin{pmatrix}0&0&1\\ 1&0&1\\ 0&1&1\end{pmatrix} Then we should have: $\hat{A} = S^{-1}AS$, and therefore $A = S\hat{A}S^{-1}$ should result in the same matrix as the first one above, but instead I get back: \begin{pmatrix}-1&0&2\\ -2&1&3\\ 0&1&1\end{pmatrix} So I'm obviously getting lost somewhere but I cannot figure out where. If anyone could point out my mistake(s) I'd appreciate it a lot.