Why does this linear map from a definition of two polynomials' resultant involve two other polynomials? Good afternoon stackexchangers.. I'm looking for more help with resultants of two polynomial functions.
In this wiki page on resultants I'm not understanding why the linear map $\varphi(P,Q) = AQ+BP$ is represented by the Sylvester matrix of $A$ and $B$. According to that definition, I've got to use a polynomial $P$ from the vector space of polynomials less than the degree of $B$ and a polynomial $Q$ from the vector space of polynomials of degree less than $A$ but I don't see  how the $P$ and $Q$ can be any polynomial from those spaces and still be represented by the same matrix.
This page also shows the matrix with just entries from the coefficients of what would be $A$ and $B$.
Here is a computation I wrote of the matrix from two polynomials and a discriminant computation.
 A: I think you're a bit confused here. The Sylvester matrix $S$ is the matrix that represents the map $(P,Q)\mapsto AQ+BP$. That is, if you give me two polynomials $P,Q$ then I can determine $AQ+BP$ by applying the matrix $S$ to the vector of coefficients of $P$ and $Q$. (There's nothing to do with the matrix representing $P$ and/or $Q$.)
Here's an example so you can better see what "representing the map $(P,Q)\mapsto AQ+BP$" means: if $A=x^3+x+1$ and $B=3x^2+2x+1$, then $$S = \begin{pmatrix} 1 & 0 & 1 & 1 & 0 \\ 0 & 1 & 0 & 1 & 1 \\ 3 & 2 & 1 & 0 & 0 \\ 0 & 3 & 2 & 1 & 0 \\  0 & 0 & 3 & 2 & 1 \end{pmatrix}$$ so if we have polynomials $Q=ax+b$ in $\mathcal{P}_{<\deg B}$ and $P=cx^2+dx+e$ in $\mathcal{P}_{<\deg A}$ then $$\begin{pmatrix} a \\ b \\ c \\ d \\ e \end{pmatrix}^T\cdot\begin{pmatrix} 1 & 0 & 1 & 1 & 0 \\ 0 & 1 & 0 & 1 & 1 \\ 3 & 2 & 1 & 0 & 0 \\ 0 & 3 & 2 & 1 & 0 \\  0 & 0 & 3 & 2 & 1 \end{pmatrix}= \begin{pmatrix} a+3c \\ b+2c+3d \\ a+c+2d+3e \\ a+b+d+2e \\ b+e \end{pmatrix}^T$$ which exactly matches what we get from computing $AQ+BP$: $$(a+3c)x^4+(b+2c+3d)x^3+(a+c+2d+3e)x^2+(a+b+d+2e)x+(b+e).$$
