For notational convenience I want to introduce a short-hand notation: Let $A\in\mathbb R^{n\times(k+1)}$ and $x\in\mathbb R^k$. My "product" should represent the affine-transformation $A\bullet x = Sx + a_0$ (where $a_0$ is the first column of $A$ and $S$ is the submatrix of $A$ when the first column $a_0$ is delted). Does this make sense? Are there any problems with this definition? Is there maybe already a similar concept I can build on?
edit: some background: My function looks like this: $$f(x) = A\left[\begin{array}{c} g\left(B\left[\begin{array}{c} h(x) \\ 1\end{array}\right]\right) \\ 1 \end{array}\right]$$ Using the notation I could simply write $f(x) = A\bullet g(B\bullet h(x))$.