# $\mathbb{R}(A/B)$ $=$ $\mathbb{R}\left(\frac{\alpha A+\beta B}{\gamma A + \delta B}\right)$, if $\alpha \delta − \beta \gamma \neq 0$

Think this question is easy but I'm having much trouble to resolve it.

Let $\mathbb{R}(A/B)$ be the field of the racional funcions generated by $\frac{A}{B}$ over $\mathbb{R}$, where $A$ and $B$ are polynomials in $\mathbb{R}$. Then for all $\alpha,$ $\beta$, $\gamma$, $\delta$, satisfying $\alpha \delta − \beta \gamma \neq 0$, we have $\mathbb{R}(A/B)$ $=$ $\mathbb{R}\left(\frac{\alpha A+\beta B}{\gamma A + \delta B}\right)$

Can someone help me or at least give a hint?

Hint: Write $A/B$ in the form \begin{bmatrix}A\\B\end{bmatrix} and now you have $$\begin{bmatrix}\alpha&\beta\\ \gamma&\delta\end{bmatrix}\begin{bmatrix}A\\B\end{bmatrix}=\begin{bmatrix}\alpha A+\beta B\\ \gamma A+\delta B\end{bmatrix}$$
• Nice hint, but how I transform the racional funtion $\frac{A}{B}$ in the vector $\left[\begin{array}{l}A\\ B \end{array}\right]$? – Matheus Manzatto Jul 14 '17 at 2:06
• We are just using the matrix notation for convenience. What is really going on behind the scenes is that we are solving a linear equation. Once you invert the matrix, you will have a recipe for $A/B$ in terms of $A'/B'$ where $A'=\alpha A+\beta B$ and $B'=\gamma A +\delta B$. – Ravi Jul 14 '17 at 2:09