5.1-05 Consequences of the axioms

Derive the basic identities that make vector calculations reliable. Rather than taking familiar cancellation rules for granted, this set asks you to connect zero, additive inverses and scalar multiplication to the axioms that justify them.

What you will practise

• Prove uniqueness of the additive identity.
• Explain additive and scalar cancellation.
• Distinguish scalar zero from vector zero.

Preparation and approach

Before starting, review the vector-space axioms and elementary equation manipulation. 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.

Division by a vector is not an available vector-space operation; cancellation must use an allowed inverse. 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 consequences of the axioms with the surrounding chapter and to revisit the reasoning after an initial attempt.