# Thread: Linear Mapping Rank Proofs

1. ## Linear Mapping Rank Proofs

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.

2. What is the nullity of an invertible map?

3. Thanks, sorry I solved this but forgot to edit it afterwards. Since the nullity of an invertible map is 0, the rank nullity theorem presents a simple solution.