6.3-07 Complementary projections

Complementary projections

Complementary projectors split a vector into its fitted component and perpendicular residual. Practice connects the identity-minus-projector construction with left null spaces, dimension counts, energy calculations and reflections.

What you will practise

• Construct and apply a complementary projector.
• Locate its range and kernel.
• Check mutual annihilation and reconstruction.
• Relate complementary components to reflection.

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 idempotence, symmetry and orthogonal decomposition. Keep a written calculation or short justification for each response, especially when several statements can be correct. This microlearning supports interpreting residuals and Householder geometry within Module 6 of the Complete Linear Algebra Course.