3. Finding a regular transformation

Let Tr_(A,A') be given by generators $(X_i,Y_i),\ i=1,\ldots,r$. We are searching for linear combinations $X(\underline{f})=\sum_{i=1}^r f_iX_i$ and $Y(\underline{f})=\sum_{i=1}^r f_iY_i$ such that $det(X(\underline{f}))\not= 0$ and $det(Y(\underline{f}))\not= 0$ simultaniously.