3.1-06 Refuting linearity

Refuting linearity — linear algebra practice

Develop short, decisive counterexamples for translations, products, squares, norms, trigonometric functions and piecewise rules. Contrast a single failed universal identity with successful but inconclusive sample tests. Basis-value agreement and zero preservation are examined as common sources of false proofs.

What you will practise

• Choose an efficient counterexample
• Diagnose missing proof assumptions
• Explain why a passing test can be insufficient

This microlearning belongs to Module 3 of the Complete Linear Algebra Course. Single-select and multiselect questions progress from interpretation to calculation, application and error diagnosis. Each problem is self-contained, and the explanations show how to justify a result rather than only identify a choice. In multiselect tasks, assess each statement separately; the number of correct choices varies.

Before starting, review the two defining linearity identities. Work through the reasoning on paper, compare your method with the explanation and revisit the assumptions behind any missed item. These skills prepare you to judge transformation claims before using matrix methods.