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When $A\vec{x}=\vec{b}$ has no exact solution, the least-squares solution $\vec{x}^*$ is defined as the vector that minimizes the Euclidean norm of the residual, $\|\vec{b} - A\vec{x}^*\|$. Geometrically, this means choosing $\vec{x}^*$ such that $A\vec{x}^*$ is the closest possible vector to $\vec{b}$ within the column space $C(A)$.

Conditions: The system $A\vec{x}=\vec{b}$ is inconsistent (no exact solution exists); Distance is measured by the standard Euclidean norm

When $A\vec{x}=\vec{b}$ has no exact solution, the least-squares solution $\vec{x}^*$ is defined as the vector that minimizes the Euclidean norm of the residual, $\|\vec{b} - A\vec{x}^*\|$. Geometrically, this means choosing $\vec{x}^*$ such that $A\vec{x}^*$ is the closest possible vector to $\vec{b}$ within the column space $C(A)$.

Conditions: The system $A\vec{x}=\vec{b}$ is inconsistent (no exact solution exists); Distance is measured by the standard Euclidean norm