Writing a Linear Map in a particular basis I've been asked to write the Linear Map $T: V \to V$ given by $Tf(x) = xf'(x)$ in the basis $\{1, x, x^2\}$. Where $V$ is the vector space of polynomials of degree no greater than $2$. 
I'm not really sure what the question is asking me to do, and I'm confused how to do this when I'm not sure what f is.
Any help would be greatly appreciated, thanks
 A: Compute $Tf$ where $f$ is each of the functions in the basis:
$$\begin{align}T(1)&=0\\
T(x)&=x\\
T(x^2)&=2x^2.
\end{align}$$
Then express each of these in terms of the basis
$$\begin{align}0&=0+0x+0x^2\\
x&=0+1x+0x^2\\
2x^2&=0+0x+2x^2.
\end{align}$$
Then you can read off the matrix
$$\begin{pmatrix}0&0&0\\
0&1&0\\
0&0&2\\\end{pmatrix}.$$
To put it another way, the $(i,j)$th entry of the matrix is the coefficient of the $j$th basis element when you express $Tf$ in terms of the basis, where $f$ is the $i$th basis element. 
A: You should write a formula for the function using the basis. That is, compute
$$T(a+bx+cx^2)$$
and write the result as a linear combination of $\{1,x,x^2\}$.
A: For $f(x)=a_0+a_1x+a_2x^2$ we have:
$$
T(f(x))=xf'(x)=x(a_1+2a_2x)=a_1x+2a_2x^2
$$
so, representing the polynomials as vectors in the basis $\{1,x,x^2\}$, the transformation is:
$$
T\begin{bmatrix}
a_0\\a_1\\a_2
\end{bmatrix}=
\begin{bmatrix}
0\\a_1\\2a_2
\end{bmatrix}
$$
and a simple inspection show that $T$ is represented by the matrix
$$
T=\begin{bmatrix}
0&0&0\\0&1&0\\0&0&2
\end{bmatrix}
$$
