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The linear least-squares problem is to choose $x\in {\mathbb{R}}^n$ minimizing the Euclidean norm of the residual:For a QR decomposition $A=QR$, orthogonality of $Q$ gives $\Vert Ax-b{\Vert }_2=\Vert Rx-Q^{T}b{\Vert }_2$. If $A$ has full column rank and $R=\left(\genfrac{}{}{0ex}{}{R_1}{0}\right)$ is in standard form with invertible upper-triangular $R_1\in {\mathbb{R}}^{n\times n}$, the minimizer solves $R_{1}x=(Q^{T}b{)}_{1:n}$.

Applying the Gram-Schmidt process to the columns of the given matrix yieldsIndeed, $Q$ is an orthogonal matrix and $A=QR$. Moreover,Back substitution in the leading triangular system givesso the unique least-squares solution isThe unused final transformed residual component is $1$, so the minimum residual norm is $1$.

For $u\in U$, define $\varphi (u)\in U^{\ast }$ by $\varphi (u)(u^{\prime })=⟨u,u^{\prime }⟩$. If $\varphi (u)=0$, then in particularso $u=0$. Thus $\varphi$ is injective. Since a finite-dimensional vector space and its dual space have the same dimension, $\varphi$ is also surjective and hence an isomorphism. This is the finite-dimensional real case of the Riesz representation theorem.

The adjoint operator of $\alpha :V\to W$ is the unique linear map ${\alpha }^{\ast }:W\to V$ satisfyingfor every $v\in V$ and $w\in W$. In the stated orthonormal bases, if $v$ and $w$ denote coordinate columns, thenTherefore the matrix of ${\alpha }^{\ast }$ is the matrix transpose $A^T$.

For $w\in W$,Hence $\ker {\alpha }^{\ast }=(\mathrm{im}\alpha {)}^{\perp }$. Taking orthogonal complements in the finite-dimensional space $W$ proves the image-kernel orthogonality for an adjoint

Put $r=\alpha (v_0)-w_0$. For any $h\in V$,This is minimized at $h=0$ exactly when $r\perp \mathrm{im}\alpha$, equivalently when ${\alpha }^{\ast }r=0$. Thus the normal equation for a linear inverse problem is

For the given linear least-squares problem,The normal equations $2x+y=3$ and $x+2y=5$ have the unique solution