3.2-01 Why basis images determine a map

Why basis images determine a map — linear algebra practice

Connect a vector’s basis expansion to the corresponding combination of its images. Work with standard and nonstandard input bases, recover unknown basis images from sums, and distinguish complete information from partial or inconsistent data. The linearity assumption remains explicit throughout.

What you will practise

• Use basis coefficients to predict outputs
• Recover images from combined inputs
• Check consistency and uniqueness of defining data

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 basis expansions and linearity. 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 build a standard matrix correctly.