3.1-01 Inputs, outputs and preimages

Inputs, outputs and preimages — linear algebra practice

Read transformations as input-output rules with a declared domain and codomain. Distinguish the whole output space from the image actually reached, then work backward from a target to its preimage. The exercises include unique, missing and parameterized families of preimages, so the vocabulary is connected to solving equations.

What you will practise

• Identify domain, codomain and image
• Evaluate coordinate formulas
• Describe empty and infinite preimages

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 coordinate vectors and basic linear equations. 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 reason about which outputs are reachable.