6.1-05 Unique orthogonal decomposition

Unique orthogonal decomposition

Unique orthogonal decomposition separates a vector into a component in a subspace and a perpendicular remainder. The questions connect explicit decompositions with the proof of uniqueness, energy identities and the nearest-point interpretation.

What you will practise

• Compute coordinate and noncoordinate decompositions.
• Check every condition of a proposed split.
• Use squared-length identities.
• Distinguish orthogonal decomposition from a general direct sum.

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 complements, dot products and basic subspace arguments. Keep a written calculation or short justification for each response, especially when several statements can be correct. This microlearning supports projection matrices and closest-point problems within Module 6 of the Complete Linear Algebra Course.