6.5-07 Householder reflections

Householder reflections

Householder reflections eliminate components while preserving geometric information. The set develops the rank-one formula, tests fixed and reversed directions, performs a numerical example and explains reflection ordering and stable sign choices.

What you will practise

• Construct and apply a Householder reflector.
• Distinguish reflection from projection.
• Use reflectors to introduce zeros.
• Track products and numerical cancellation choices.

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 orthogonal projectors, matrices and norms. Keep a written calculation or short justification for each response, especially when several statements can be correct. This microlearning supports Householder QR and stable matrix factorization within Module 6 of the Complete Linear Algebra Course.