Least squares solves $\min_x\|Ax-b\|_2$. A minimizer always exists in finite dimensions, and its fitted vector $A\hat x$ is the orthogonal projection of $b$ onto $\operatorname{Col}A$. The residual is orthogonal to all columns, yielding $A^*(A\hat x-b)=0$. Coefficients are unique exactly when $A$ has full column rank.
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- Definition of least-squares solution to a linear systemA least-squares solution is any $\hat x$ minimizing $\ Ax-b\ 2$ over all domain vectors. Equivalently, it solves the...
- Definition of least-squares error of a linear systemThe residual for a candidate $x$ is $r=b-Ax$. Its Euclidean error is $\ r\ 2$; the sum of squared errors is $\ r\...
- Formula for computing the least squares solution to a linear system.If $A$ has full column rank, the least-squares solution is $\hat x=(A^ A)^{-1}A^ b$, obtained from the normal...
- Formula for computing the least squares solution to a linear systemWith reduced QR $A=QR$ and full column rank, solve the triangular system $R\hat x=Q^ b$. With rank deficiency, SVD...
- The least squares solution to a linear system is unique if and only if the columns of the coefficient matrix are linearly independent.Least-squares coefficients are unique iff $\ker A=\{0\}$, equivalently the columns are independent. If $z\in\ker A$,...
- Definition of the least-squares linear fit to 2-dimensional dataFor real data $(x i,y i)$, a least-squares line $y=a+bx$ minimizes $\sum i(y i-a-bx i)^2$. The design matrix has rows...
- Formula for the least-squares linear fit to 2-dimensional dataIf $S {xx}=\sum i(x i-\bar x)^2>0$, the fitted slope is $b=\sum i(x i-\bar x)(y i-\bar y)/S {xx}$ and $a=\bar y-b\bar...
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