least-squares solution

$\hat{\mathbf x}$ — the $\mathbf x$ minimizing $\lVert A\mathbf x - \mathbf b\rVert^2$ .

When $A\mathbf x=\mathbf b$ has no exact solution, a least-squares solution $\hat{\mathbf x}$ minimizes $\lVert A\mathbf x-\mathbf b\rVert^2$ — the sum of squared residuals — making $A\hat{\mathbf x}$ as close to $\mathbf b$ as any choice of $\mathbf x$ can. If $\mathbf b$ happens to lie in $\operatorname{Col}(A)$ after all, the minimum is $0$ and $\hat{\mathbf x}$ is an ordinary exact solution.