3.6-02 Independence, span and unique solutions

Independence, span and unique solutions — linear algebra practice

Link homogeneous uniqueness, independent columns, full column span and target solvability in square systems. Contrast consistency for one target with uniqueness for one target, and use standard basis targets to reason about every possible right-hand side.

What you will practise

• Translate between column and equation statements
• Distinguish existence from uniqueness
• Use square dimensions in basis arguments

This microlearning belongs to Module 3 of the Complete Linear Algebra Course. Single-select and multiselect questions progress from interpretation to calculation, application and error diagnosis. Each problem is self-contained, and the explanations show how to justify a result rather than only identify a choice. In multiselect tasks, assess each statement separately; the number of correct choices varies.

Before starting, review pivots, column span and homogeneous systems. Work through the reasoning on paper, compare your method with the explanation and revisit the assumptions behind any missed item. These skills prepare you to recognize equivalent descriptions of a bijective map.