Equivalence of Definitions of Basis of Vector Space

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Theorem

Let $K$ be a division ring.

Let $\struct {G, +_G, \circ}_K$ be an vector space over $K$.


The following definitions of the concept of Basis of Vector Space are equivalent:

Definition 1

A basis of $G$ is a linearly independent subset of $G$ which is a generator for $G$.

Definition 2

A basis is a maximal linearly independent subset of $G$.


Proof