3.6-03 A chain of equivalences

A chain of equivalences — linear algebra practice

Build short proofs connecting inverse identities, injectivity, surjectivity and transpose invertibility. Choose legitimate implication steps, use contrapositives and identify the missing link in an equivalence cycle. The questions emphasize why the connections hold, not just recognition of a theorem list.

What you will practise

• Prove selected implications
• Transfer invertibility to the transpose
• Complete logical equivalence cycles

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 the invertible matrix theorem and inverse identities. 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 write concise structural proofs.