I have a question about expressing some coplanar vectors in terms of a smaller number of orthogonal basis vectors.
I have a small number of vectors $P$ in a higher dimensional space $N$, something like $5$ vectors in a $100$ dimensional space. I'm trying to figure out how to find
- the vector from the origin that is orthogonal to the 4-D plane that contains all $5$ points,
- $4$ orthogonal vectors parallel to that plane, and
- the $P\times P$ matrix to express the $5$ original vectors in terms of these new $5$ basis vectors.
This is problem I've been trying to crack for a work project and implement in Python, but I'm not very good at matrix algebra. I've been doing the 3Blue1Brown series and trying to figure it out myself.