Consider an $m\times n$ matrix over a field $K$, $A\in M(m, n, K)$. The Moore-Penrose pseudoinverse of $A$ is a matrix $A^+\in M(n, m, K)$ such that,
For $x\in M(m, 1, K)$, let $A|x$ denote the matrix obtained by appending $x$ after the last column of $A$. Given two column matrices $u,v\in M(m, 1, K)$, is there a clear relationship between $(A|u)^+$ and $(A|v)^+$?
I realize the notion of "clear relationship" is vague, but any sort of algebraic identity relating to two, or evidence of no such identity would be extremely useful. I have not found literature in either case. Thank you!