Consider chain of linear maps between finitely dimensional vector spaces $E$, $F$ and $G$ over $\mathbb{Z}/2$:
$E\xrightarrow{A}F\xrightarrow{B}G$
then we take transpose $A^t$ and $B^t$ and consider chain
$E\xleftarrow{A^t}F\xleftarrow{B^t}G$.
Is there any way to prove that $\ker(B)/Im(A)\cong \ker(A^t)/Im(B^t)$?
Note: This question is regarding isomorphism of homologies in proof of Poincare duality for Morse Homology, however I need purely algebraic result here. In the proof maps $A$ and $B$ are given by chain maps for Morse function $f$ while $A^t$ and $B^t$ correspond to chain maps between morse complexes of function $-f$.