6.4-06 Uniqueness and rank deficiency

Uniqueness and rank deficiency

A unique best fitted vector can coexist with infinitely many coefficient vectors. This microlearning develops that distinction using duplicate columns, null-space families, minimum-norm selection and near-dependence examples.

What you will practise

• Identify the correct uniqueness criterion.
• Describe all coefficient minimizers.
• Separate exact dependence from sensitivity.
• Preserve fitted outputs when removing redundant columns.

Begin with recognition and interpretation, then work through calculations, error diagnosis and more demanding reasoning. Single-select and multi-select questions ask you to evaluate the stated assumptions as well as the result. Worked explanations connect the answer to a method or a counterexample, so you can check why tempting alternatives fail.

Before you start

Review rank–nullity and the closest-point theorem. Keep a written calculation or short justification for each response, especially when several statements can be correct. This microlearning supports reliable interpretation of rank-deficient regression within Module 6 of the Complete Linear Algebra Course.