Filling the ambient space

Filling the ambient space: linear algebra practice

Ask whether a collection can reach every vector in its surrounding space. These questions use geometry and explicit combinations to test complete reachability, including examples where several listed vectors still miss a direction. The emphasis is on constructive explanations instead of determinant or row-reduction tests.

What you will practise

• Decide whether available directions fill two- or three-dimensional space
• Construct arbitrary-target representations
• Distinguish the number of listed vectors from the number of usable directions

How to use this questionset

Prerequisites: Linear combinations, spans of one and two vectors, and Cartesian coordinates. 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.

A vector count alone does not guarantee coverage. Check whether the supplied directions actually allow all target coordinates. Full-space reachability provides an intuitive starting point for basis and invertibility later in the course. This microlearning belongs to Module 1, Vectors and Geometry, in the Complete Linear Algebra Course.