Matrix Representation of a Linear Transformation I'm not quite sure how to approach the following question.

Let $T:\Bbb R_2[x] \rightarrow \Bbb R_2[x]$ be a linear transformation which
  satisfies:
$ \begin{cases} T(2x+1) = 2x^2+1 \\ T(x+2) = x^2-3x+2\\ T(x^2-x+2) =
 -5x+3 \end{cases} $
$W = \{x^2,x^2-2x,x^2-x+2\}$ is a basis for $\Bbb R_2[x]$.
Find the matrix $[T]_\mathit{W}^\mathit{W}$

 A: As you’re being asked to find the matrix of a linear transformation, a somewhat different approach to that in the comments above is to move to $\mathbb R^3$ right off the bat, since you’re going to end up there eventually.  
Recall that the columns of the matrix of a linear transformation are the images of the basis. The three vectors for which $T$ is defined in the problem are clearly linearly independent, so we’ll use them as our ordered basis $B=(2x+1,x+2,x^2-x+2)$. Let $E$ be the standard basis $(1,x,x^2)$. We can immediately write down the matrix relative to these bases: $$[T]_E^B=\begin{bmatrix}1&2&3\\0&-3&-5\\2&1&0\end{bmatrix}.$$ (I’m not sure which convention you’re using for the matrix notation. By $[T]_E^B$ I mean that $B$ is the “input” basis and $E$ the “output” basis of the matrix, so that we can write $[\mathbf v]_W=[id]_W^V[\mathbf v]_V$ for a change of basis—upper and lower indices “cancel.”)  
To obtain $[T]_W^W$ from this we need to perform a few changes of basis: $$[T]_W^W=[id]_W^E [T]_E^B [id]_B^W = [id]_W^E [T]_E^B [id]_B^E [id]_E^W.\tag1$$ The matrix $[id]_E^W$ that maps from $W$ to the standard basis has the elements of $W$ for its columns; $[id]_W^E$ is the inverse of this matrix. Similarly, $[Id]_E^B$ is the inverse of the matrix that has the elements of $B$ as its columns. Putting this all together, equation (1) becomes $$[T]_W^W=\begin{bmatrix}0&0&2\\0&-2&-1\\1&1&1\end{bmatrix}^{-1}\begin{bmatrix}1&2&3\\0&-3&-5\\2&1&0\end{bmatrix}\begin{bmatrix}1&2&2\\2&1&-1\\0&0&1\end{bmatrix}^{-1}\begin{bmatrix}0&0&2\\0&-2&-1\\1&1&1\end{bmatrix}.$$ I’ll leave the rest of the computation to you.  
I should mention that this approach isn’t really different from that suggested in the comments to the question. When you’re constructing the various change-of-basis matrices above, you’re computing the coefficients of linear combinations of basis vectors that produce other basis vectors. It’s only the mechanism of doing so that’s different.
