Existence of a Basis of a Vector Space

Overview

The existence of a basis for an arbitrary vector space has been shown using Zorns lemma. (see alabiso)

Dependent and Independent Vectors

For a set of vectors {% S \subset V %} of the vector space {% V %}, the following are equivalent

• {% S %} is a maximal linear independent set
• {% S %} is a minimal spanning set
• {% S %} is a basis

Theorem - Existence of a Basis

Every linearly independent set {% S \subset V %} can be extended to a maximal linearly independent set.