5.3-03 Unique representation

Connect the two basis conditions to existence and uniqueness of representations. This short set combines coefficient calculations with subtraction proofs and a redundant spanning list whose representations reveal a free parameter.

What you will practise

• Identify which property supplies existence.
• Prove uniqueness by subtracting expansions.
• Track changes caused by reordering a basis.

Preparation and approach

Before starting, review spanning, independence and two-variable equations. 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 spanning list can supply many representations of the same vector when it contains redundancy. 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 unique representation with the surrounding chapter and to revisit the reasoning after an initial attempt.