Proofs about Matrix Rank I'm trying to prove the following two statements. I can prove them easily by considering the matrix as a representation of a linear map with a given basis, but I don't know a proof which uses just the properties of matrices.  
First, I want to prove that similar matrices have the same rank. This seems obvious because the rank is just the dimension of the image of the linear map, but these matrices represent the same map (just in a different basis).  
Next, I want to prove that $rank(AB)\le \min(rank(A),rank(B))$. Again, this seems relatively obvious if we just consider $AB$ as the composition of the two maps , but I can't see how to do it with matrices.

If $n$  vectors $v_1,..v_n$ are linearly independent(dependent) then for non singular matrix $P$ the vectors $Pv_1,...Pv_n$ also  will be independent(dependent). From this fact and from the definition of the rank as a number of linearly independent columns (rows) we immediately can conclude that similar matrix have the same rank.  The second fact follows from  the dimension left and right kernels. 
