Is a least squares solution to $Ax=b$ necessarily unique 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.
 A: The statement needs a crisp definition. Consider the matrix $\mathbf{A}\in\mathbb{C}^{m\times n}_{\rho}$ ($m$ rows, $n$ columns, rank $\rho$).
Uniqueness
If the data vector $b$ is not in the null space $\mathcal{N}(\mathbf{A}^{*})$ the least squares solution exists. If the number of columns is greater or equal to the rank of $\mathbf{A}$, $n\ge \rho$, the solution is unique.
Demonstration
The linear system $\mathbf{A}x=b$,
$$
\begin{bmatrix}1 & 0\end{bmatrix}
\begin{bmatrix}x \\ y\end{bmatrix} =
\begin{bmatrix}b\end{bmatrix},
$$
offers the least squares solution
$$
x_{LS} = \color{blue}{x_{particular}}
 + \color{red}{x_{homogeneous}} = 
\color{blue}{\begin{bmatrix}b \\ 0\end{bmatrix}} + 
\alpha \color{red}{\begin{bmatrix}0 \\ 1\end{bmatrix}}
$$
where the arbitrary constant $\alpha \in \mathbb{C}$
The particular solution lives in the range space $\color{blue}{\mathcal{R}(\mathbf{A})}$. The homogeneous solution inhabits the null space $\color{red}{\mathcal{N}(\mathbf{A}^{*})}$. When this null space is trivial (when $n\ge \rho$), the is no homogeneous contribution and the least squares solution is unique.
Unique least square solutions
A: If $b$ is in the column space of $A$ then a true solution exists, which is unique if and only if the columns of $A$ are linearly independent.
If $b$ is not in the column space of $A$ then the least-squares solution is the orthogonal projection of $b$ to the column space of $A$, which is of course unique.
