5.2-05 Sums of subspaces

Interpret the sum of two subspaces as all combinations of one vector from each. Work with spans and coordinate equations to see how new directions arise and why this construction produces the smallest subspace containing both inputs.

What you will practise

• Build spanning lists for sums.
• Check membership by solving for components.
• Use containment and outside directions to identify the resulting space.

Preparation and approach

Before starting, review spanning sets and the subspace test. 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.

The sum of subspaces usually contains more vectors than their set-theoretic union. 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 sums of subspaces with the surrounding chapter and to revisit the reasoning after an initial attempt.