Adjoint of linear operators

Question: Let $\displaystyle T$ be a linear operator on a finite-dimensional vector space $\displaystyle V$. Show that:

1. $\displaystyle N(T^*T)=N(T)$

2. $\displaystyle rank(T) = rank(T^*)$

3. $\displaystyle rank(A^*A)=rank(AA^*)=rank(A)$

I figured that if I can prove (1), then it follows via the dimension theorem that $\displaystyle rank(T^*T)=rank(TT^*)=rank(T)$. Then if I just find a basis, I can use the previous fact to solve (3). I just cannot get started on (1), though. And what about (2)? Any tips would be helpful - thanks!