We define the row/column space of a matrix $A$ to be all possible sums of rows/columns of $A$.
The row space ($V(A)$) and column space ($W(A)$) form a lattice.
See example matrix and respective row space lattice here:
$$A=\begin{bmatrix}1&0&1&0\\0&1&0&1&\\0&0&1&1&\\0&1&0&0\end{bmatrix}\to \langle V(A);\land,\lor\rangle,$$ where $\langle V(A);\land,\lor\rangle=$
and the example matrix and respective column space lattice here: $$\begin{bmatrix}1&0&1&0\\0&1&0&1\\0&0&1&1\\0&1&0&0\end{bmatrix}\to \langle W(A);\land,\lor\rangle,$$ where $\langle W(A);\land,\lor\rangle=$
All row and column space lattices are inverses of each other. Any ideas on a proof to this?