3.1-03 Homogeneity

Homogeneity — linear algebra practice

Explore how output changes when an input vector is scaled. Negative and zero scalars are as important as positive ones. Examples separate real homogeneity from positive-scalar behavior and include a map that preserves all scalar multiples yet fails addition, clarifying why the two linearity requirements must be checked separately.

What you will practise

• Test positive, negative and zero scalars
• Recognize linear and nonlinear scaling behavior
• Separate homogeneity from additivity

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 scalar multiplication and coordinate formulas. 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 evaluate both conditions needed for linearity.