Linear maps of polynomials, the bases of the space and their corresponding matrix. Suppose $T \in \mathrm{Hom}(\mathscr{P}_3(\mathbb{R}),\mathscr{P}_4(\mathbb{R}))$ is defined by:
$$Tp(x)=(x^2p(x))',$$
for all $x \in \mathbb{R}$ and $S \in\mathrm{Hom} (\mathscr{P}_4(\mathbb{R}),\mathscr{P}_3(\mathbb{R}))$ is defined by:
$$Sp(x)=xp''(x),$$
for all $x \in \mathbb{R}$. Let $(1,x,x^2,x^3)$ be a basis for $\mathscr{P}_3(\mathbb{R})$ and $(1,x,x^2,x^3,x^4)$ be a basis for $\mathscr{P}_4(\mathbb{R})$. Find $ST$ and $\mathscr{M}(ST)$ (i.e. the matrix for ST).
I could use some help in solving this problem. I have tried a number of things which all seem to lead me nowhere. I understand that the linear map will operate on $p(x)$ but how do I incorporate the respective basis of the space? and were do I go from there?
 A: $STp=(x^2(xp''(x)))'=(x^3p''(x))'$
Computing it for the generic polinomial in $\mathscr{P_4}(\mathbb{R})$
$ST(c_0+c_1x+c_2x^2+c_3x^3+c_4x^4)=c_2x^2+c_3x^3+c_4x^4$, where $c_i\in\mathbb{R}, i=0, 1, 2, 3, 4$
Having chosen $(1, x, x^2, x^3, x^4)$ for the ordered base in $\mathscr{P_4}$, you get
$\mathscr{M}(ST)=
\begin{bmatrix}
0&0&0&0&0\\
0&0&0&0&0\\
0&0&1&0&0\\
0&0&0&1&0\\
0&0&0&0&1
\end{bmatrix}$
A: The map $ST$ is just the composition of the two transformations, so you can find an expression for it by substituting the the value $Tp$ for $p$ in the expression for $Sp$.  
As for finding its matrix, there are a couple of ways you could proceed. If you simplify the expression that you got for $S(Tp)$, then you can pretty much read it off from the coefficients of the result of applying this map to a generic third-degree polynomial. Remember that each column of the matrix will be the image of the corresponding basis vector. For example, if $ST(a+bx+cx^2+dx^3)=5ax^2+10bx^3$ (I’m just making up numbers here), then the matrix would be $$
\pmatrix{0&0&0&0\\0&0&0&0\\5&0&0&0\\0&10&0&0},
$$ which you can see by setting each coefficient in the generic polynomial to $1$ in turn while setting the rest to zero.
Another way is to work out the individual matrices for $S$ and $T$ and multiply them. It’s worth doing it both ways to check your work.
