Linear Transformations between 2 non-standard basis of Polynomials 
If 
  $$ 
A = \begin{pmatrix}  1 & -1 & 2 \\
-2 & 1 &-1 \\ 1 & 2 & 3  \end{pmatrix} 
$$ 
  is the matrix representation of a
  linear transformation $T : P_3(x) \to P_3(x)$  with respect to
  bases $\{1-x,x(1-x),x(1+x)\}$ and $\{1,1+x,1+x^2\}$. Find T.

While i have worked with transforming non-standard to standard basis, this is the first one i am encountering with transformation between 2 non-standard polynomial basis. I am not sure if i am working out rightly.
$T[1-x]    = 1(1) -2(1+x) +1(1+x^2)$
$T[x(1-x)] = -1(1) +1(1+x) +2(1+x^2)$
$T[x(1+x)] =  2(1) -1(1+x) +3(1+x^2)$
Therefore, $T[a(1-x)+b(x(1-x))+c(x(1+x))] = (a-b+2c)(1) + (-2a+b-c)(1+x) +(a+2b+3c)(1+x^2)$
Is this fine ?

Everything you've written is correct, although I suspect the problem is asking you to find $T[a + bx + cx^2]$.  
