6.5-01 One orthogonalization step

One orthogonalization step

One orthogonalization step removes the part of a vector already represented by earlier orthonormal directions. Practice follows the subtraction carefully, checks the remaining direction and handles normalization and nonunit basis vectors.

What you will practise

• Subtract one or several orthogonal components.
• Calculate residuals and normalize them.
• Explain why the partial span is preserved.
• Handle already perpendicular and zero-residual cases.

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 projection and normalization. Keep a written calculation or short justification for each response, especially when several statements can be correct. This microlearning supports the complete Gram–Schmidt algorithm within Module 6 of the Complete Linear Algebra Course.