This question might be so elementary , but I love to see some geometric/algebraic approach for these facts concerning transpose matrix. I know some basic thing about DUAL space but those are not satisfying in the way I know.
$1. \: rank (A)=rank(A^t)$
$2. \: ker (A) = co-image(A^t)$
$3. \: co-ker(A)= image(A^t)$
P.S. One may re-state the third fact as
$$Ax=0 , x^ty=0 \implies \exists z: A^tz=y$$
Is there any simple way just using basic algebra to prove the fact?
Thanks community