QR decomposition ($A=QR$)

A reduced QR decomposition of a full-column-rank matrix writes $A=QR$, where $Q$ has orthonormal columns and $R$ is upper triangular with positive diagonal.

For a full-column-rank $n\times k$ matrix $V$, extend the thin QR decomposition $V=Q_1{R}_1$ to an orthogonal matrix $Q=(Q_1{Q}_2)$. Thenso $Q^T$ maps the column space of $V$ onto the first $k$ coordinate directions. Successive Householder transformations construct the same reduction without first forming a full orthogonal basis.

A linear least-squares problem minimizes $\Vert Ax-b{\Vert }_2$ over $x$. If $A=QR$ has full column rank, orthogonality reduces it to the upper-triangular system formed from the leading block of $R$ and the leading entries of $Q^{T}b$.