Find a matrix transformation mapping $\{(1,1,1),(0,1,0),(1,0,2)\}$ to $\{(1,1,1),(0,1,0),(1,0,1)\}$ 
Find a matrix transformation mapping $\{(1,1,1),(0,1,0),(1,0,2)\}$ to $\{(1,1,1),(0,1,0),(1,0,1)\}$.

Is the answer 
$$
\begin{bmatrix}1& 0& -1\\0& 1& 1\\0& 0& 1\end{bmatrix}?
$$ 
I understand the concept of Matrix Transformation, I don't think I'm doing it right.

The columns of the matrix tell you where it sends the standard basis vectors. For instance if I am interested in the third column then I need to determine what the action of our linear operator is on the column vector ,
$$\left[ \begin{array}{c}0 \\ 0 \\ 1 \end{array}\right].$$
This vector can be written as a linear combination of the vectors  used to define the linear operator,
$$\left[ \begin{array}{c}0 \\ 0 \\ 1 \end{array}\right] = 
\left[ \begin{array}{c}1 \\ 0 \\ 2 \end{array}\right]
-\left[ \begin{array}{c}0 \\ 1 \\ 0 \end{array}\right]
-\left[ \begin{array}{c}1 \\ 1 \\ 1 \end{array}\right].$$
Multiplying both sides by our linear operator $M$ we get, 
$$M\left[ \begin{array}{c}0 \\ 0 \\ 1 \end{array}\right] = 
M\left[ \begin{array}{c}1 \\ 0 \\ 2 \end{array}\right]
-M\left[ \begin{array}{c}0 \\ 1 \\ 0 \end{array}\right]
-M\left[ \begin{array}{c}1 \\ 1 \\ 1 \end{array}\right].$$
Note that we know what $M$ does to the vectors on the right so we can just substitute those values in and add,
$$M\left[ \begin{array}{c}0 \\ 0 \\ 1 \end{array}\right] = 
\left[ \begin{array}{c}1 \\ 0 \\ 1 \end{array}\right]
-\left[ \begin{array}{c}0 \\ 1 \\ 0 \end{array}\right]
-\left[ \begin{array}{c}1 \\ 1 \\ 1 \end{array}\right]
=\left[ \begin{array}{c}0 \\ -2 \\ 0 \end{array}\right].$$
The resulting vector is the third column of our matrix,
$$ M = \left[\begin{array}{ccc} 
\  & \  & 0 \\
\  & \  & -2 \\
\ & \  & 0 
\end{array}\right].$$
A similar process will yield the other collumns.
