6.3-06 The general projection matrix

The general projection matrix

General subspace projection can be computed from any independent column basis. These questions follow the inverse-Gram formula, simplify it for orthonormal columns and test how changing the basis affects the resulting closest point.

What you will practise

• Construct correctly sized projection matrices.
• Project onto coordinate and noncoordinate planes.
• Compare different bases for the same span.
• Handle redundant columns without using an invalid inverse.

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 normal equations and Gram-matrix invertibility. Keep a written calculation or short justification for each response, especially when several statements can be correct. This microlearning supports basis-independent interpretation of least-squares fitted values within Module 6 of the Complete Linear Algebra Course.