Reachability and span

Reachability and span: linear algebra practice

Explore span as a reachability question: can the available directions produce a target using real weights? You will supply a combination when a target is reachable and use a coordinate constraint when it is not. The questions also distinguish a span from a segment, ray or region obtained by restricting coefficients.

What you will practise

• Interpret span as all reachable real linear combinations
• Test a target using a constructive combination or invariant
• Distinguish full spans from restricted coefficient regions

How to use this questionset

Prerequisites: Linear combinations and coordinate arithmetic. 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.

Unless stated otherwise, span permits arbitrary real coefficients, including negative and fractional values. This reachability viewpoint will support geometric independence and later column-space descriptions of systems. This microlearning belongs to Module 1, Vectors and Geometry, in the Complete Linear Algebra Course.