Span of one vector

Span of one vector: linear algebra practice

Examine the complete set generated by one vector. These questions separate the generating arrow from the line of all its real multiples, include the special zero case, and test whether alternative generators give the same set. Short membership checks connect the geometric description to coordinate ratios.

What you will practise

• Describe all real multiples of one nonzero vector
• Handle the zero-generator exception
• Compare generators of the same line

How to use this questionset

Prerequisites: Real scalar multiplication and the definition of span. 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 span is a set of vectors, not one long arrow or a bounded segment. Its geometric description must include the allowed coefficient range. The one-direction case is the foundation for lines and for detecting redundant directions. This microlearning belongs to Module 1, Vectors and Geometry, in the Complete Linear Algebra Course.