6.6-07 Building orthogonal polynomials

Building orthogonal polynomials

Build low-degree orthogonal polynomials by applying Gram–Schmidt to monomials. Practice uses the unweighted integral product on the symmetric unit interval and distinguishes monic, endpoint and unit-norm normalizations.

What you will practise

• Construct constant, linear and quadratic orthogonal directions.
• Calculate projection coefficients and squared norms.
• Relate the results to low-degree Legendre polynomials.
• Extend the construction to a cubic and explain dependence on the inner product.

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 Gram–Schmidt and polynomial product integrals. Keep a written calculation or short justification for each response, especially when several statements can be correct. This microlearning supports best polynomial approximation in an orthogonal basis within Module 6 of the Complete Linear Algebra Course.