# rank and nullity

• Nov 22nd 2010, 03:10 AM
alexandrabel90
rank and nullity
given that g:U--> V and f: V-->W where g and f are linear map,

which is always true and which might not be always true:

1.rank (f.g) less than or equal to rank(g)

2.rank(f.g) greater than or equal to rank (f)

3.nullity(f.g) grater than or equal to nullity(f)

4.nullity (f.g) less than or equal to nullity(g)

how do you go about doing such qns...
• Nov 22nd 2010, 05:02 AM
HallsofIvy
You do such questions by thinking about what "rank" and "nullity" mean. If f:V-->W, then f(V) is some subspace of W. The rank of f is the dimension of that subspace. Of course, the dimension of f(V) cannot be larger than the dimension of V itself.

f.g(U)= f(g(V)). The rank of f.g is the dimension of that space and, as I said, it cannot be larger than the dimension of g(V) which [b]is[b] the rank of g.

If f:V-->W then the nullity of f is the dimension of its "null space"- the subspace of V such that f(v)= V whenever v is in that subspace. In particular, if g(v)= 0, then f(g(v))= f(0)= 0 so the null space of g is a subspace of the null space of f.g. What does that tell you about their dimensions?
• Nov 22nd 2010, 05:06 AM
alexandrabel90
Nullity of g is smaller than nullity of fg.

Nullity of f is smaller than the nullity of fg too right
• Nov 22nd 2010, 04:48 PM
topspin1617
Huh...

Are you sure you have the choices written down correctly? I don't think any of the choices in the original post are necessarily true...

What is true is:

• $\mathrm{null}(g)\leq \mathrm{null}(f\circ g)$
• $\mathrm{rank}(f\circ g)\leq \mathrm{rank}(f)$