The assignment was to find the transposed of the following function, but I approached it using matrices as I was more familiar with that. I think the point of the assignment was to understand transposed function when we're not necessarily talking about matrices, but here's what I did.
Let $\int:P_2(\mathbb{R})\to P_3(\mathbb{R})$ be defined by $p(x)\mapsto \int_0^x p(x)dx$. What does the transposed function $\int^T$ do to arbitrary elements?
I rewrote them in a vector representation - for example: $3x^2+2x+8 \implies \begin{pmatrix}3\\2\\8\end{pmatrix}$
and found the following matrix:
$$[\smallint]_\beta=A= \begin{pmatrix} \frac{1}{3} & 0 & 0 \\ 0 & \frac{1}{2} & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix} $$
Where $\beta=(1,x,x^2)$ or $((1,0,0)^T,(0,1,0)^T,(0,0,1)^T)$. $A$ maps exactly the vector representations from $P_2(\mathbb{R})$ to the vector representations of their integrals in $P_3(\mathbb{R}).$ Then:
$$A^T= \begin{pmatrix} \frac{1}{3} & 0 & 0 & 0\\ 0 & \frac{1}{2} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ \end{pmatrix} $$
This defines the function $\int^T:P_3(\mathbb{R})\to P_2(\mathbb{R})$, and looking again at what this matrix does, I found the following function: $ax^3+bx^2+cx+d\mapsto \frac{a}{3}x^2+\frac{b}{2}x+c$.
In the end I'm not sure if it's correct, I'd expected something like a derivative or something you could write at once using $p(x)\mapsto$ something. Using the matrices might also not be the most elegant, or even correct way.
Does anyone have any tips on how to do this using just the functions and no matrices? Is this what the end result must be?