6.5-05 Least squares with QR

Least squares with QR

Solve least-squares problems using orthonormal coordinates and a triangular system. This set follows reduced and full QR viewpoints, computes fitted vectors and residual errors, and identifies the rank assumptions required for a unique solve.

What you will practise

• Form the QR least-squares right-hand side.
• Solve R by back substitution.
• Recover fitted outputs and residual norms.
• Compare QR with exact normal-equation solutions.

Begin with recognition and interpretation, then work through calculations, error diagnosis and more demanding reasoning. Single-select and multi-select questions ask you to evaluate the stated assumptions as well as the result. Worked explanations connect the answer to a method or a counterexample, so you can check why tempting alternatives fail.

Before you start

Review reduced QR and ordinary least squares. Keep a written calculation or short justification for each response, especially when several statements can be correct. This microlearning supports solving fits without explicitly forming the Gram matrix within Module 6 of the Complete Linear Algebra Course.