Suppose I have a real square invertible matrix $A$ of size $n\times n$.
Consider the space $$S_1=\{Ax:x\in\Bbb R^n\}$$ and $$S_2=\{A'x:x\in\Bbb R^n\}$$ where $'$ is transpose.
Assuming that we have $B\neq A$ and is invertible.
How many pairs of $(Ax,x)$ and $(Bx,x)$ do I need to know to tell infact $$B=A'$$ holds true?
The problem is to find to out using the structure of the space the transformations are transposes of each other without looking at $A,B$ themselves. Consider this as a black box model in computation.