QR factorization separates the transformation by A into an orthogonal part Q and an upper-triangular part R. For a consistent system Ax = b, left-multiplying by Qᵀ removes the rotational part and reduces the problem to solving Rx = Qᵀb. For inconsistent systems, QR gives the same best solution as normal equations while preserving the geometry more cleanly.
QR factorization turns a general system into one whose essential deformation is encoded in an upper-triangular matrix.
The chapter compares these viewpoints directly and shows how the QR method isolates distortion in R while keeping the orthogonal part separate.
It removes the orthogonal part of the transformation by A. Since Q preserves lengths and angles, multiplying by Qᵀ undoes the rotation or reflection and leaves only the triangular deformation R to solve.
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Because QR solves the same least-squares problem without explicitly forming AᵀA. It keeps the orthogonal part separate, reduces the problem to back-substitution in R and avoids distorting both sides of the equation as much as the normal-equations route does.