Constructing dependent and independent sets

Constructing dependent and independent sets: linear algebra practice

Choose or construct vector collections meeting stated requirements. These problems combine span and independence: a set can reach the desired space while containing redundant vectors, or be independent without reaching a larger surrounding space. Small examples and impossibility arguments help separate these ideas clearly.

What you will practise

• Build sets with a requested span and independence status
• Detect impossible size and geometry requests
• Construct counterexamples to overgeneralized dependence claims

How to use this questionset

Prerequisites: Spans, homogeneous relations and dependence in two and three dimensions. The questions progress from direct interpretation to application and reasoning. Work out an answer before comparing the choices; for multi-select questions, evaluate every statement independently. Worked explanations show the reasoning and address relevant misconceptions, so an incorrect attempt can guide your next review.

State the ambient space and the required span before counting vectors. Distinctness, nonzero entries and independence are different properties. Constructing examples will strengthen later reasoning about bases, dimension and minimal generating sets. This microlearning belongs to Module 1, Vectors and Geometry, in the Complete Linear Algebra Course.