6.2-02 Coordinates by inner products

Coordinates by inner products

An orthonormal basis makes coordinates accessible through inner products. Practice translates between vectors and coordinate lists, explains the required normalization, and distinguishes a complete basis from an incomplete orthonormal family.

What you will practise

• Recover signed coordinates in rotated bases.
• Reconstruct vectors from coordinates.
• Adjust coefficients for nonunit orthogonal bases.
• Use orthonormal coordinates for norms and dot products.

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 bases, normalization and dot products. Keep a written calculation or short justification for each response, especially when several statements can be correct. This microlearning supports efficient projections and matrix representations in orthonormal coordinates within Module 6 of the Complete Linear Algebra Course.