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.