Krylov (the guy the subspaces are named after) was a naval engineer. Cholesky was an artillery officer.
"Fire control computers ... solve ... fire control problems."
https://www.youtube.com/watch?v=gwf5mAlI7Ug
U.S. NAVY BASIC MECHANISMS OF FIRE CONTROL COMPUTERS MECHANICAL COMPUTER INSTRUCTIONAL FILM 27794
The starting step is one that is impossible to misunderstand. From there, go one concrete step to another.