3.6-04 Limits of the theorem

Limits of the theorem — linear algebra practice

Test which familiar equivalences survive for rectangular matrices. Explicit embeddings and projections show the difference between one-sided inverses and a two-sided inverse. Changing the codomain to the image provides a final comparison between a coordinate-space map and its natural restricted range.

What you will practise

• Diagnose misuse of square assumptions
• Interpret left and right inverses
• Separate full row from full column conditions

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 rectangular maps, dimensions and the square theorem. 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 apply structural results with correct hypotheses.