5.2-06 Direct sums

Explore direct sums through the uniqueness of component decompositions. Examples involving lines, polynomial terms and several summands connect the definition to intersections, proof arguments and the limitations of pairwise tests.

What you will practise

• Solve for components in a two-subspace decomposition.
• Use intersections to test directness.
• Diagnose a three-summand example with nonunique decomposition.

Preparation and approach

Before starting, review subspace sums, independence and solving small linear systems. 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.

Directness does not require orthogonality, and pairwise zero intersections are insufficient for several summands. 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 direct sums with the surrounding chapter and to revisit the reasoning after an initial attempt.