I know there always exists a least-square solution $\hat{x}$, regardless of the properties of the matrix $A$. However, I keep finding online that least-square can have infinitely many solutions, if $A$ is not full column rank.
Shouldn't $\hat{x}$ always be unique, as the minimization of a quadratic function (the error) always yields a global minima/maxima? Therefore, regardless of what the matrix $A$ is (even if it is a badly constructed matrix with dependent columns), least-square should find a single 'best' solution $\hat{x}$?
Is there an easy (or intuitive) proof showing why would the least-square method produce infinitely many solutions if there are dependent columns?