6.3-03 Residual orthogonality

Residual orthogonality

Residual orthogonality is the geometric condition behind least squares. This set connects the nearest point in a column space with matrix equations, directional changes in the objective and a proof of global optimality.

What you will practise

• Locate residuals in observation space.
• Check perpendicularity to every design column.
• Derive the first-order condition by expansion.
• Distinguish sufficient tests from weaker one-vector checks.

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 subspace projection and matrix–vector multiplication. Keep a written calculation or short justification for each response, especially when several statements can be correct. This microlearning supports deriving and verifying the normal equations within Module 6 of the Complete Linear Algebra Course.