I was going through Les Valiant's seminal paper and I had a tough time with Proposition 4.3 on page 10 of the paper.
I cannot see why is it the case that if there is a generator with certain values for $valG$ with a basis $\{(a_1,b_1) \ldots (a_r,b_r)\}$, then there exists some generator with same $valG$ values for any basis $\{(xa_1,yb_1) \ldots (xa_r,yb_r)\}$ ($1^{st} kind$) or $\{(xb_1,ya_1) \ldots (xb_r,ya_r) \}$ ($2^{nd} kind$) for any $x,y \in F$.
Valiant points out the reason in preceding paragraph - namely the $1^{st}$ kind of transformation can be achieved by appending to every input or output node an edge of weight $1$. The $2^{nd}$ kind of transformation, Valiant says, can be achieved by appending to input or output nodes chains of length $2$ weighted by $x$ and $y$ respectively.
I have not been really able to understand these statements. Maybe they are already clear, but still I cannot really see why the above construct helps achieve any realizable $valG$ values with one basis with the new basis which is one of the above types.
Please help illuminate them to me. On a different note, are there some tensor free surveys for hologaphic algorithms available online. Most of them use tensors which, sadly, scare me :-(
Best -Akash