how do you prove that rank(A) = rank (A^t)?
im trying to states that Ax=0, A^tAx=O then nullspace of A = nullspace of A^t.
then rank(A) = rank (A^t)..but this wont work if A is not a sq matrix right?
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.