6.6-01 Inner-product axioms

Inner-product axioms

Inner-product axioms extend perpendicularity beyond ordinary coordinate dot products. These questions test proposed real products, build counterexamples and connect positive definiteness with injective linear transformations.

What you will practise

• Check bilinearity, symmetry and positive definiteness.
• Use explicit vectors to disprove a candidate.
• Calculate through inner-product linearity.
• Identify valid sums, scalings and transformed 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 real vector spaces, bilinear expressions and independence. Keep a written calculation or short justification for each response, especially when several statements can be correct. This microlearning supports geometric reasoning in weighted, matrix and function spaces within Module 6 of the Complete Linear Algebra Course.