What is the nullity of an invertible map?
I have the following problem:
Let U, V, W be finite dimensional vector spaces over R and let
L : V -> U and M : U -> W be linear mappings
a) Prove rank(M(L)) <= rank(M)
b) Prove rank(M(L)) <= rank(L)
c) Prove that if M is invertible, then rank(M(L)) = rank(L)
Now...I know that rank(N) = dim(Range(N)) for some linear mapping N
I can use this to prove part a, since range(M(L)) must be some subset of range(M) and therefore dim(range(M(L))) <= dim(range(M)) and rank(M(L)) <= rank(M).
I don't know how I am supposed to prove b though, is there a way to show that dim(range(M(L))) <= dim(range(L))
edit: I think I can use the rank nullity theorem to show that since:
rank(M(L(V))) + nullity(M(L(V))) = dim (L(V))
rank(M(L(V))) <= dim(L(V))
and since L(V) is, by definition, the range of L:
rank(M(L)) <= rank(L)
I am however, still stuck on how to go about proving c...
I also don't know how to go about proving c, is there some related property based on the invertibility of M?
Any help would be greatly appreciated.