6.5-03 Dependence during Gram–Schmidt

Dependence during Gram–Schmidt

A zero Gram–Schmidt residual means a direction was already represented. This set distinguishes exact redundancy from a small floating-point residual and explains how to continue constructing a basis without inventing new directions.

What you will practise

• Recognize dependent inputs and count retained directions.
• Skip exact zero residuals correctly.
• Analyze a near-dependent parameter example.
• Use scale-aware language for numerical rank decisions.

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 rank. Keep a written calculation or short justification for each response, especially when several statements can be correct. This microlearning supports rank-aware orthogonalization and numerical methods within Module 6 of the Complete Linear Algebra Course.