Need some help with this... doesn't seem to be a decent example in my book. Appreciate any help!
1) Apply the transformation T to each and every element of E and write the result as a linear combination of E itself;
2) Take the transpose , if you apply transformations from the left and vectors from the right), or directly (otherwise) the coefficients matrix of the above: this is $\displaystyle [T]_E$.
For example:
$\displaystyle T(1)=1+1 = 2=2\cdot 1+0\cdot x+0\cdot x^2+0\cdot x^3$
$\displaystyle T(x)=x+x-2=-2+2x=(-2)\cdot 1+2\cdot x +0\cdot x^2+0\cdot x^3$ ...
So the first two columns of $\displaystyle [T]_E$ are $\displaystyle \begin{pmatrix}2\\0\\0\\0\end{pmatrix}\,,\,\,\begi n{pmatrix}\!\!-2\\2\\0\\0\end{pmatrix}$ (or the first two rows if you write the map on the right of the vector).
Tonio