Specifically, say I have a linear map $\mathbb{R}^2\rightarrow\mathbb{R}^2$. I want to construct from it a map between a regrouping of the vector spaces. Concretely, I start with \begin{align} \left[\begin{array}{c} x' \\ y' \end{array}\right] &= \left[\begin{array}{cc} a & b \\ c & d \end{array}\right]\left[\begin{array}{c} x \\ y \end{array}\right]. \end{align}
Has the process of finding the map that this implies from $(y,y')$ to $(x,x')$ been studied? Does it have a name?
Constructing the map is straightforward. One way to do this is to break the matrix up into equations, invert one of them, and reconstruct the matrix relationship. Inverting the bottom equation gives: $$ x = c^{-1} y' - d c^{-1}y. $$ This changes the top equation to: $$ x' = ac^{-1} y' + (b - adc^{-1})y, $$ thus: \begin{align} \left[\begin{array}{c} x \\ x' \end{array}\right] &= \left[\begin{array}{cc} -dc^{-1} & c^{-1} \\ b - adc^{-1} & ac^{-1} \end{array}\right]\left[\begin{array}{c} y \\ y' \end{array}\right]. \end{align}
I think the most fitting name for this process would be transverse, since it is similar, visually, to transposing the inputs and outputs of the equation. Obviously, the necessary and sufficient condition for this transverse to exist is for $c\neq0$.