Writing the change of coordinate matrix for bases I need help with writing the change of coordinate matrix from B to C, and from C to B where
$B =  \begin{bmatrix}
    1\\
    2 \\  
    1 \\ 
\end{bmatrix}$ , $\begin{bmatrix}
    1\\
    1 \\  
    1 \\ 
\end{bmatrix} $, $\begin{bmatrix}
    2\\
    1 \\  
    1 \\ 
\end{bmatrix} $
and 
$C =  \begin{bmatrix}
    1\\
    1 \\  
    -1 \\ 
\end{bmatrix}$ , $\begin{bmatrix}
    1\\
    -1 \\  
    1 \\ 
\end{bmatrix} $, $\begin{bmatrix}
    -1\\
    1 \\  
    1 \\ 
\end{bmatrix} $
are two bases in $\mathbb{R}^3$.
what I tried:
I think that the change of basis matrix is just $C$. From there, I think that I should find the vector relative to $C$, but I have not been able to make any progress. 
 A: $T_{S\leftarrow B}\mathbf x_B = \begin{bmatrix} 1&1&2\\2&1&1\\1&1&1\end{bmatrix} \mathbf x_B$
Would transform a vector $\mathbf x$ in basis $B$ into the standard basis.
And 
$T_{S\leftarrow C} = \begin{bmatrix} 1&1&-1\\1&-1&1\\-1&1&1\end{bmatrix} \mathbf x_C$
Transforms a vector $\mathbf x$ in basis $C$ into the standard basis.
$T_{S\leftarrow C}^{-1} = T_{C\leftarrow S}$ would then covert a vector from the standard basis to basis C.
If you want to convert a vector from basis B to basis C.
$T_{S\leftarrow C}^{-1}T_{S\leftarrow B}\mathbf x_B = T_{C\leftarrow S}T_{S\leftarrow B}\mathbf x_B$
Will convert $\mathbf x_B$ from $B$ to standard and then from standard to $C.$
A: First of all, make sure you understand what a change of coordinate matrix is. It must take a vector from the first basis and send it to another vector from the second basis. 
So, if $M$ is the change of coordinate matrix, $\{\vec b_1, \vec b_2, \vec b_3\}$ your first basis (which you call $B$) and $\{\vec c_1, \vec c_2, \vec c_3\}$ your second basis (which you call $C$), you must have:
$$M\vec b_i=\vec c_i$$
for $i=1,2,3$.
Finally, as the formula suggests, inverting the matrix $M$ will give you the change of coordinate matrix from $\{\vec c_1, \vec c_2, \vec c_3\}$ to  $\{\vec b_1, \vec b_2, \vec b_3\}$
