6.5-04 Constructing QR

Constructing QR

QR factorization records Gram–Schmidt coordinates in an upper triangular matrix. Practice covers reduced factor shapes, coefficient entries, reconstruction, diagonal signs, column spans and the convention that makes the factors unique.

What you will practise

• Construct and verify reduced QR factors.
• Explain why R is upper triangular.
• Recover columns and basis coordinates.
• Handle sign changes and positive-diagonal conventions.

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 Gram–Schmidt and matrix products. Keep a written calculation or short justification for each response, especially when several statements can be correct. This microlearning supports triangular least-squares solves within Module 6 of the Complete Linear Algebra Course.