Could someone please provide a detailed proof for the follow

Could someone please provide a detailed proof for the following: It is a fact that for any overdetermined matrix A, the nullspace of A and the nullspace of A^TA are the same. You do not have to explain this fact. Based on the equality of the nullspaces of A and A^TA, explain why an overdetermined system Ax=b has a unique least squares solution if A is full rank. Thank you very much!

Solution

it has a unique least square solution because the overdetermined system AX=B having full rank is only proven by it.

the least square solution is given by the uniqueness of the determinent of a matrix. It does not has a detailed proof. it is only proved by the non zero determinent of the given matrix A.