If we view (linear) operators as nothing than multipliers multiply their eigenvectors with numbers
then aren't they the same with projections from a finite point in projective geometry?
There are projection operators in functional analysis projecting the whole space into its subspace, but they're not linear, while this projection is analogous to the definition of linear operator and it stretches the whole space. The Möbius transform is linear transform on homogeneous coordinates in $\Bbb C^2$.
Can I see linear operators as protections in projective geometry without trouble?