5.3-02 Verifying a basis

Verify a basis by checking both sufficient coverage and absence of redundancy. Polynomials, matrices, planes and function spans provide varied settings for calculations, counterexamples and parameter-dependent basis decisions.

What you will practise

• Separate spanning from independence.
• Check candidates belong to the stated space.
• Solve a parameter case and justify a function-space basis.

Preparation and approach

Before starting, review span, independence and elementary 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.

Matching the expected number of candidates is not enough unless membership and a basis condition are established. 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 verifying a basis with the surrounding chapter and to revisit the reasoning after an initial attempt.