Consider an $n \times n_1$ matrix $A_1$ and an $n \times n_2$ matrix $A_2$ with the following properties:
- $\mathrm{Rank} (A_1)=n_1$,
- $\mathrm{Rank} (A_2)=n_2$,
- $\mathrm{Rank} (A_1 : A_2)=n_1+n_2$
Let $B=(B_1:B_2)$ be a nonsingular $n \times n$ matrix such that $A_1^TB_1=I_{n_1}$ and $A_1^TB_2=0$.
The paper I'm reading says that
- $\mathrm{Rank} (A_2^TB)=n_2$,
- $\mathrm{Rank} (A_2^TB_2)=n_2$,
- $\mathrm{Rank} (A_1^TB:A_2^TB)=n_1+n_2$
I'm not sure why the above are true. Can someone help me understand? I think part of the reason is that $B$ is nonsingular but I'm not sure.
Hints:
(i) For the first use the fact that $B$ is invertible.
(ii) For the second multiply $(A_1:A_2)^T$ with $B$ and use $A_1^TB_2=0$.
(iii) The last follows from the argument you write in your above comment.