3.1-04 Proving linearity

Proving linearity — linear algebra practice

Build proofs that cover arbitrary vectors and real scalars. Connect coordinate formulas, matrix multiplication and polynomial evaluation with the combined linearity identity. Domain assumptions, parameter restrictions and proof gaps receive attention alongside routine examples, so recognizing a plausible formula becomes a justified mathematical claim.

What you will practise

• Use arbitrary-input arguments
• Identify valid vector-space domains
• Prove linearity of formulas and combinations of maps

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 additivity, homogeneity and basic algebra. 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 construct matrices for verified linear maps.