The least squares solution to a linear system is unique if and only if the columns of the coefficient matrix are linearly independent.

Created almost 9 years ago, updated about 8 hours ago

Least-squares coefficients are unique iff $\ker A=\{0\}$, equivalently the columns are independent. If $z\in\ker A$, then $A(x+z)=Ax$, so any nonzero null vector causes nonuniqueness. Conversely, two minimizers have the same unique projected fitted vector and differ by a null vector.

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