if S : U-->V and T : V---> W are linear maps, and U,V and W are vector spaces over the same field K, how can we prove the following:

1) rank (TS) < or = rank (T)

2) rank (TS) < or = rank (S)

3) if U=V and S is nonsingular then Rank (TS) = Rank (T)

4) if V=W and T is nonsingular then Rank (TS) = Rank (S)