5.3-04 The exchange principle

Use the exchange principle to compare independent and spanning lists and to replace basis vectors safely. The practice develops the nonzero-coefficient argument behind replacement, equal basis sizes, extraction and extension.

What you will practise

• Choose valid basis replacements.
• Compare the sizes of independent and spanning lists.
• Recover original basis vectors after a replacement.

Preparation and approach

Before starting, review basis expansions and solving linear relations. The questions progress from recognition and direct applications to error diagnosis, justification and transfer. Single-select items ask for one defensible answer; multiselect items require you to evaluate every statement independently.

A replacement is justified by its coefficient in a basis expansion, not simply by the new vector being nonzero. Work through the calculation or argument before checking the explanation, then use the feedback to identify the step or assumption that needs attention.

This microlearning forms part of Module 5, Vector Spaces and the Four Fundamental Subspaces, in the Complete Linear Algebra Course. Use it to connect the exchange principle with the surrounding chapter and to revisit the reasoning after an initial attempt.