Change of basis? So the question is...
A transformation $T$ is denoted by $T(x,y)=(x+y,x-y)$.
$C$ is the basis $\{(1,-1),(1,1)\}$
$D$ is the basis $\{(1,2),(1,0)\}$
I know $T(C)=\{(0,2),(2,0)\}$
But how do I express that in terms of $D$ and more importantly what would the matrix representation of $ [T]_C^D $ look like?
Some formatting help would also be appreciated. Thanks!
 A: I won't do the computation but I will give you something much better -- the recipe of how to find what you are looking for: 
Note that in the standard basis $(1,0), (0,1)$ the matrix of $T$ is
$$ M = \begin{array}{cc} 1 & 1 \\ 1 & - 1 \end{array}$$
This is a map taking a vector represented in the standard basis to a vector represented in the standard basis. 
What you want is a matrix that takes a vector in the basis $C$ into a vector in the basis $D$. So given a vector in the basis $C$ you could transform it into a vector in the standard basis, apply $T$ and transform it into the basis $D$.
If $M_C$ denotes the matrix transforming from $C$ into the standard basis and $M_D$ from the standard basis into $D$ then the matrix you want will be $M_DMM_C$. 
Now all that's left is to compute $M_C$ and $M_D$. But that's easy: From $C$ into the standard basis will be the matrix with columns $(1,-1), (1,1)$ (that's the basis vectors of $C$). For the other matrix take the inverse transform from $D$ into the standard basis and compute its inverse. 
