I have to show following things and have no idea what I have to do at b)
a) $GL(n,K) \times K^{n \times m} \rightarrow K^{n \times m}: (g,A) \mapsto gA$ is a group action.
b) The Rank is an invariant of this group action.
c) Rank is a separating invariant (Proof/Counter Example)
a) With the identity matrix $E_n \in GL(n,K)$ follows: $E_n \cdot A = A$ for all $A \in K^{n \times m}$. And since matrix multiplication is associative, $(gh)m = g(hm)$ for all $g,h \in GL(n,K)$ and all $m \in K^{n \times m}$.
How do I show, that the rank is an invariant?