I have a problem I am stuck at.
Suppose object $a$ in a vector space is represented by a subspace, i.e., a set of orthonormal vectors $(u_1, u_2, \ldots, u_n)$. I want to represent $a$ as a single vector. How should I be finding the weights,
$a = \alpha_1 u_1 + \alpha_2 u_2 + \ldots + \alpha_n u_n$
How to find these alpha? Any standard method?
I think if I can find a vector for $a$ close enough to compare then there must be a method that tries to minimize the loss between $a$ and its reconstruction in terms of linear combination.