5.2-04 Intersections and unions

Understand how intersections and unions behave differently for subspaces. Geometric examples lead into closure proofs, containment conditions and the key subtraction argument used to diagnose a union that does not stay closed under addition.

What you will practise

• Compute intersections of coordinate planes.
• Distinguish nested unions from nonnested unions.
• Explain closure for an arbitrary intersection of subspaces.

Preparation and approach

Before starting, review set membership, subspace closure and elementary geometry. 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 union can be a subspace when one component contains the other; an automatic rejection misses that case. 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 intersections and unions with the surrounding chapter and to revisit the reasoning after an initial attempt.