Matrix Representation Problem 2 Can someone help me solve this Problem here? 


And what is the monomial Basis, is it just a polynomial of degree 2 or less? 
And for example in order to calculate $L_{W_1 \leftarrow V_1}$,should I firstly put the Basis of $V_1$ in $L$ and then calculate it with respect to the Basis of $W_1$ (which should just give the matrix itsself,as $W_1$ is the standard Basis)? 

 A: From reading the problem i think $\mathcal{P}_2$ denotes the $\mathbb{R}$-vectorspace of polynomials with real coefficents of degree $\leq 2$. Lets take $p = a_2 X^2 +a_1 X +a_0$  in $\mathcal{P}_2$. Then the monomial basis $V_1$ is the familiy $(X^0 =1, X, X^2)$, as you can see $p$ is a linear combination of the basis elements. 
Lets compute the matrix representation $L_{W_1 \leftarrow V_1}$. As you have said, we need to plug in the basis $V_1$ into our given map, recall that it is enough for a homomorphism of vector spaces to evaluate on basis elements, so 
$L(X^0)= \begin{bmatrix} X^0(1) + 2 X^0(1) & 4({X^0})^{\prime}(0)\\
3X^0(2) - ({X^0})^{\prime \prime}(0) & ({X^0})^{\prime \prime}(1)+ ({X^0})^{\prime}(1)
\end{bmatrix} = \begin{bmatrix} 3 & 0 \\
3 &  0 \end{bmatrix} = 3 \begin{bmatrix} 1 & 0 \\
0 &0 \end{bmatrix} + 3 \begin{bmatrix} 0 & 0 \\
1 &0 \end{bmatrix} = \\ =3 w_1 + 0 w_2 +3 w_3 + 0 w_4$
So the first column in our matrix $L_{W_1 \leftarrow V_1}$ is just $\begin{pmatrix} 3 \\  0 \\ 3 \\ 0 \end{pmatrix}$. I hope you got the concept, i will leave the rest of the computations up to you.
