I want to learn an algorithmic approach to solve the following general problem: How to find all elements in $PGL(n+1,K)$ which maps a $k$-dimensional subspace, say $U$, of $\mathbb{P}^{n}$ to another $k$-dimensional subspace, say $W$, of $\mathbb{P}^{n}$?
I guess i know how to solve this question for one element. In other words, suppose i just need to find one element of $PGL(n+1,K)$ mapping $U$ to $W$. Then, we can do the following, I guess:
1- Write $U = <p_{1},...,p_{k}>$ for some points $p_{1},...,p_{k}$
2- Write $W = <q_{1},...,q_{k}>$ for some points $q_{1},...,q_{k}$
3- Extend both lists to frames. Since $PGL(n+1,K)$ acts regularly on frames of $P^{n}$, we are done.
As a concrete example: Find all elements of $PGL(4,K)$ which maps $L_{1}$ to $L_{2}$ where $L_{1} = <(1,2,1,1),(2,3,1,0)>$ and $L_{2} = <(1,1,1,2),(3,1,0,1)>$.
Thanks, in advance.