6.5-02 The Gram–Schmidt algorithm

The Gram–Schmidt algorithm

Construct orthonormal bases using the full Gram–Schmidt process. The questions build from two-vector calculations to three-dimensional examples, partial-span invariants, order dependence and the inner-product reasoning behind the algorithm.

What you will practise

• Carry out successive projection subtractions.
• Normalize each independent residual.
• Track output order and partial spans.
• Diagnose incorrect subtraction against overlapping inputs.

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 one-step orthogonalization and linear independence. Keep a written calculation or short justification for each response, especially when several statements can be correct. This microlearning supports QR factorization and orthogonal bases for abstract spaces within Module 6 of the Complete Linear Algebra Course.