Using basis $e=[x^3,x^2,x,1]$ instead of $e=[1,x,x^2,x^3]$

It makes no sense to ask what is the matrix of a linear transformation without fixing bases. If the person to whom this question is being asked is free to choose those bases, then both answers are correct.

• First : "I'm using notation $(a,b,c,d)$ to mean $(ax^3,bx^2,cx,d)$" doesn't make any sense. You wanted to say, "I'm using the notation $(a,b,c,d)$ to denote $ax^3+bx^2+cx+d$".

• Secondly : In your notation, does for example $x$ refer to the vector $\begin{pmatrix}0\\1\\0\\ 0\end{pmatrix}$ or to the vector $\begin{pmatrix}0\\0\\1\\0\end{pmatrix}$ ? Because to use the basis $\{x^3,x^2,x,1\}$ for $$\left\{\begin{pmatrix}0\\0\\0\\1\end{pmatrix}, \begin{pmatrix}0\\0\\1\\0\end{pmatrix}, \begin{pmatrix}0\\1\\0\\0\end{pmatrix}, \begin{pmatrix}1\\0\\0\\0\end{pmatrix}\right\}$$ is correct, but using $\{x^3,x^2,x,1\}$ for $$\left\{\begin{pmatrix}1\\0\\0\\0\end{pmatrix}, \begin{pmatrix}0\\1\\0\\0\end{pmatrix}, \begin{pmatrix}0\\0\\1\\0\end{pmatrix}, \begin{pmatrix}0\\0\\0\\1\end{pmatrix}\right\}$$ is conventionally not correct but makes sense with your notation (and it's what you seem to have made).

But your work didn't deserve 0 mark. In my opinion, you deserve almost all points for this question (since you precise that $(a,b,c,d)$ refer to $ax^3+bx^2+cx+d$). Strange that you got 0 mark !