given that f:V ->W and g: U->V, why isit that the
nullity (fg) < = nullity (f) and
nullity (fg) > nullity (g)?
how do you tell in general?
suppose u is in null(g). then fg(u) = f(g(u)) = f(0) = 0. so null(fg) contains null(g), so its (null(fg)'s) dimension has to be at least as big (as null(g)). (they could be the same, for example, g could be the 0-map).
on the other hand, u in null(fg) means g(u) is in null(f)∩g(U), which is perhaps smaller than null(f) (there might be v in V with f(v) = 0, but that aren't images g(u) for some u in U).
if by dim(fg) you mean dim(fg(U)), dim(fg(U)) ≤ dim(g(U))
and dim(fg(U)) ≤ dim(f(V)), since g(U) is contained in V.
pictorally, g may shrink U, but it can't make g(U) bigger than V. fg(U) = f(g(U)), so fg may shrink U even more than g does (it gets two chances).
(remember null( ) measures how much gets shrunk to 0).
similarly fg shrinks less than f, because fg can only act on those elements of V that survive via g from U, whereas f can act on all of V (the f by itself
has a larger domain, than the f in fg).