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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

  1. the vector from the origin that is orthogonal to the 4-D plane that contains all $5$ points,
  2. $4$ orthogonal vectors parallel to that plane, and
  3. 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.

Any thoughts?

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  • $\begingroup$ There isn’t one vector orthogonal to the subspace spanned by the 5 vectors. The orthogonal subspace is at least 100-5 = 95 dimensional. It sounds like you want to use the Gram–Schmidt process to find the orthogonal bases for both subspaces. $\endgroup$ – Joe Aug 24 '20 at 22:12
  • $\begingroup$ I'll into Gram-Schmidt, thank you. Wouldn't there be only 1 vector from the origin that is orthogonal to the plane? $\endgroup$ – Eric Leeson Aug 24 '20 at 22:30
  • $\begingroup$ I may have misunderstood what you said/want. By “4-D plane”, I thought you meant the subspace spanned by the vectors. But it sounds like you want the position vector of the closest point in the affine space containing the points associated with the 5 vectors (I’m assuming they are position vectors) to the origin. Yes, that will be unique $\endgroup$ – Joe Aug 25 '20 at 3:01
  • $\begingroup$ If your 5 position vectors are $x_1,...,x_5$, then you can subtract $x_5$ from every vector, letting $v_i = x_i - x_5$ for $1 \le i \le 4$. Let $A$ be the matrix with those 4 column vectors. Then the vector you want is $$A(A^\top A)^{-1}A^\top(-x_5) + x_5$$ If you do Gram-Schmidt on the 4 vectors $v_i$, that’ll give you #2. $\endgroup$ – Joe Aug 25 '20 at 3:19
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If I understand what you are trying to do, I think this Python code will accomplish it.

import numpy as np

# dimension of linear space
N = 100

# 5 column vectors (position vectors)
x = np.random.normal(0, 1, size=(N,5))

# matrix of displacement vectors
# the affine space containing the 5 points is
# all points of the form Ac + x[:,4]
A = x[:,:4] - x[:,4:5]

# the closest point in that affine space to the origin is
y = (A @ np.linalg.inv(A.transpose() @ A) @ A.transpose() @ (- x[:,4:5]) 
     + x[:,4:5])

# this should be zero - but isn't exactly, due to numerical linear algebra
A.transpose() @ y

# QR factorization
# the columns of q are an orthogonal basis for the columns of A
q, r = np.linalg.qr(A)

# add the vector y as a column
p = np.concatenate((q,y), axis=1)

# the orginal vectors are given by the equation p c = x
c = np.linalg.inv(p.transpose() @ p) @ p.transpose() @ x

# p c - x should be zero, but here we see the norms of the column vectors are
# all around 10^-15
np.linalg.norm(p @ c - x, axis=0)
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