if A is an mxn matrix, Aᵀ is an nxm matrix.

AAᵀ is an mxm matrix, AᵀA is an nxn matrix, both these products are pefectly well-defined, but...

by the rank-nullity theroem, dim(null(A)) = n - rank(A), whereas dim(null(Aᵀ)) = m - rank(Aᵀ),

so unless m = n, the nullspaces will NOT have the same dimensions.

what you want to do is prove that if rank(A) = k, that A has k linearly independent rows, as well as k linearly independent columns.