Let $A$ be an $m$ x $n$ matrix, and suppose that $b\in\mathbb{R}^n$ is a vector that lies in the column space of $A$. Is a least squares solution to $Ax=b$ necessarily unique? If so, give a detailed proof. If not, find a counter example.
I understand that a least-squares solution to $Ax=b$ is a vector $\hat{x}\in\mathbb{R}^n$ such that $\|b-A\hat{x}\|\le\|b-Ax\|$ for any vector $x\in\mathbb{R}^n$ which gives me the impression that the least squares solution to $Ax=b$ is not necessarily unique. However, I'm at a loss as to how to prove this.