Linear transformation using non-standard basis Find a linear transformation $T:P_{2}({R})\longrightarrow P_{4}(R)$ so that 
$T(1) = x^4$
$T(x+x^2) = 1$
$T(x-x^2) = x+x^3$
I have only solved problems using standard basis, and now I have no idea on how to deal with this.

You have that
$$x = \frac{(x+x^2) + (x-x^2)}{2}$$
So 
$$T(x) = T\left( \frac{(x+x^2) + (x-x^2)}{2} \right) = \frac{T(x+x^2) + T(x-x^2)}{2}$$
$$ = \frac{1+x+x^3}{2}$$
And
$$T(x^2) = T(x+x^2-x) = T(x+x^2)-T(x)$$
$$=1 - \frac{1+x+x^3}{2} $$
