Let $A$ be a $n\times n$ invertible matrix. Suppose $$A=\begin{pmatrix}A_{11}&A_{12}\\ A_{21}&A_{22} \end{pmatrix}$$ $$A^{-1}=\begin{pmatrix}B_{11}&B_{12}\\ B_{21}&B_{22} \end{pmatrix}.$$ where $A_{11}$ is a $l\times k$ matrix, $B_{11}$ is a $k\times l$ matrix, $1<k,l<n$. How to show $$W=\{\alpha; A_{12}\alpha=0\}$$ and $$U=\{\beta; B_{12}\beta=0\}$$ have the the dimension, so is isomorphic.

Intuitively, we have to show $$(n-k)-rank A_{12}=(n-l)-rank(B_{12}).$$ But it sounds impossible at the first glance.


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