5.2-07 Dimensions of sums

Use the dimension formula for sums and intersections as both a calculation tool and a structural check. The practice includes recovering missing dimensions, enforcing ambient bounds and explaining why shared basis directions are counted only once.

What you will practise

• Calculate dimensions with overlap.
• Identify impossible subspace specifications.
• Follow a basis-extension proof of the sum formula.

Preparation and approach

Before starting, review subspace sums, intersections and finite bases. 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.

Adding dimensions without accounting for shared directions can exceed the ambient dimension. 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 dimensions of sums with the surrounding chapter and to revisit the reasoning after an initial attempt.