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**topspin1617** The first thing to do is to compute $\displaystyle \left[ T\right]_B$, the matrix for the linear transformation $\displaystyle T$ with respect to the basis $\displaystyle B$.

It will be a 3 by 3 matrix, as 3 is the dimension of our vector space. Each row and column will stand for a basis vector; lets name them $\displaystyle b_1=x-x^2,b_2=-1+x^2,b_3=-1-x+x^2$. The columns and rows of the matrix will correspond to our basis vectors; say the columns and rows correspond to $\displaystyle b_1,b_2,b_3$ respectively. We need to do the following:

Find the image of each basis vector under $\displaystyle T$. Write these images out in terms of our basis. Then we have $\displaystyle \left[ T\right]_B=(t_{ij})$; that is, **the entry in column j, row i of $\displaystyle \left[ T\right]_B$ is the coefficient of $\displaystyle b_i$ in the expression for $\displaystyle T(b_j)$**.

Let's try it for our first basis vector $\displaystyle b_1=x-x^2$:

$\displaystyle T(x-x^2)=4-4x-4x^2=-8(-1+x^2)+4(-1-x+x^2)=0b_1-8b_2+4b_3$

Thus, the first column in the matrix should be $\displaystyle \left( \begin{array}{c}0\\-8\\4\end{array}\right)$