6.5-06 Modified Gram–Schmidt

Modified Gram–Schmidt

Modified Gram–Schmidt changes when projection coefficients are computed. These questions trace the updated residual, explain exact-arithmetic equivalence with the classical form and examine the numerical purpose of reorthogonalization.

What you will practise

• Compare original-column and updated-residual calculations.
• Execute a modified orthogonalization step.
• Explain rounding and cancellation effects.
• Use reconstruction and orthogonality diagnostics.

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 classical Gram–Schmidt and basic floating-point awareness. Keep a written calculation or short justification for each response, especially when several statements can be correct. This microlearning supports judging orthogonalization quality in numerical work within Module 6 of the Complete Linear Algebra Course.