5.2-01 The subspace test

Apply the subspace test to equations, inequalities, parametric descriptions and geometric sets. The questions move from detecting a missing zero to justifying closure and diagnosing cases that pass one part of the test but fail another.

What you will practise

• Check zero, addition and real scalar closure.
• Rewrite a parametric set as a span.
• Choose explicit counterexamples for nonlinear or one-sided constraints.

Preparation and approach

Before starting, review vector-space operations and basic coordinate geometry. 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.

Containing zero and being closed under one operation are not enough to establish a subspace. 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 the subspace test with the surrounding chapter and to revisit the reasoning after an initial attempt.