1. ## nullity

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?

2. 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).

3. just wondering, would i be able to tell if the dim (fg) is greater than or equal or less than or equal to the dim(g) and the dim(f)?

4. 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).