A is invertible real function,x is a vector
on what law
I presume you mean that A is a linear transformation- whether it is invertible is not really relevant. This is just the definition of : If A is a linear transformation from vector space U to vector space V, then is defined as the linear transformation from V back to U such that for any u in U and any v in V. The first inner product is in V and the second in U.
Starting from with and that gives