Exercise-4

LADR

Exercise 1

The empty set is not a vector space. The empty set fails to satisfy only one of the requirements listed in the definition of a vector space. Which one?

Solution

The existence of the additive identity. Every vector space should have at least one element, namely the additive identity, or the zero element. This is the only property that deals with existence of elements in a vector space. Rest of the properties are vacuously true for an empty set.

Footnotes

  1. Exercise-4, Exercises 1B, Page-16, Linear Algebra Done Right, Fourth Edition, Sheldon Axler↩︎