@trenta3 one post of mine that talks about psuedo inverse and span: math.stackexchange.com/a/317053/43193, just look for the section proof

@trenta3 I remember I might have some perinent posts on my math.stackexchange. I haven't posted too terribly much so it may be of use to you to look. Otherwise, if I think of what I was talking about here, I may be able to explain better later. no source-- I think I was doing it on my own really

I only just realized that maybe it is ok to go all the way and take the single term $0^{\underline{0}}=1$ which would then give the formula for $d=0$. It is just a weird concept to consider taking a factor of $1$ out and leaving $0$ behind.

I thought as much myself, but have not seen it in that book. Note that this formula actually eradicates the power on $x$, as opposed to just changing the basis, even when not applying any difference.

If $\mathbf A$ is symmetric and so is the update to it, then I get that the Sherman-Morrison formula works as is (replacing inverse with pseudo-inverse of course). Otherwise, if I am correct, the formula gives you only a general inverse, and correction using the null space is required to make it the desired pseudo-inverse.