If you consider a linear map $f : \mathbb{R}^{2} \to \mathbb{R}^{3}$ given by $$ f(x, y) = (2x, 3y, x + y) $$ Which can be represented like so: $$ A = \begin{bmatrix} 2 & 0 \\ 0 & 3 \\ 1 & 1 \\ \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} $$
where it is super intuitive that we are multiplying rows by columns, thats how we can find the polynomial version of matrix multiplication.
How can I interpret the presence of powers in a linear mapping in this example? $f : P_3(\mathbb{R}) \to \mathbb{R}^{2}$ defined by: $$ f(a + bx + cx^2 + dx^3) = \begin{bmatrix} a+d \\ b+c \end{bmatrix} $$
is there a way for me to convert it into matrix polynomial function (like above the notation is interchangeable)?
Of course, a matrix representation depends on the bases chosen for $P_3(\mathbf R)$ and $\mathbf R^2$. For both vector spaces, I suppose they're the canonical basis of each: $$(1, x,x^2,x^3)\quad\text{and}\quad e_1=\begin{bmatrix}1\\0\end{bmatrix}, e_2=\begin{bmatrix}0\\1\end{bmatrix}.$$ W.r.t. these bases, the columns of the matrix are the coordinates of the images of the vectors in basis $(1, x,x^2,x^3)$: $$ A=\begin{bmatrix}1&0&0&1\\0&1&1&0\end{bmatrix} .$$