I have a problem from my past paper I can't figure the logic to, even after seeing the answers.
The question goes
【Q】Let $V=\mathbb{R}[X]_{<4}$ be the vector space of real polynomials of degree less than 4. $U=\mathbb{R}[X]_{<1}$ is a subspace of V. Define the differential mapping $D:\mathbb{R}[X]_{<4} \to \mathbb{R}[X]_{<4}$. Such a $D$ gives rise to the mapping $\delta:\mathbb{R}[X]_{<4}/U \to \mathbb{R}[X]_{<4}/U$. Using the basis $({X+V, X^2+V, X^3+V})$ of $\mathbb{R}[X]_{<4}$, what is the matrix representing $\delta$ in terms of the said basis?
【My attempt】I can construct the $\delta$ with the given mapping and the canonical mapping $\mathbb{R}[X]_{<4} \to \mathbb{R}[X]_{<4}/U$ so that bit I am convinced with.
Then, for some $P \in \mathbb{R}[X]_{<4}$ this can be represented by $P=c_1(X+V)+c_2(X^2+V)+c_3(X^3+V)$ and I can put that in a matrix form. Basically, what I am trying to do is, if I can construct an equation of the form $Bk=Ac$ where $A,B$ are the matrices that transforms the standard basis of $\mathbb{R}[X]_{<4}$ to the given basis, and $c,k$ are coefficient matrices, then I can inverse $B$ to find $B^{-1}A$ which is the matrix I want.
This logic just comes from my experience of linear transforms and its representing matrices.
【Question】 However, I am not entirely sure if this is the right method for this and, actually, so far, I haven't gotten anywhere...My main question is, $\delta$ takes $P+U \to D(P)+U$ so I am not sure if I am to use the same basis for both $P+U$ and $D(P)+U$. Should I differentiate each basis for the latter??
Can anyone kindly give me some extensive explanation? I really appreciate some help here, would be great if someone can enlighten the path! Thanks a lot in advance