Generator of Vector Space is Basis iff Cardinality equals Dimension

Theorem

Let $E$ be a vector space of $n$ dimensions.

Let $G$ be a generator for $E$:

$G$ is a basis for $E$ if and only if $\card G = n$.

Proof

Necessary Condition

Let $G$ be a basis for $E$.

From Cardinality of Basis of Vector Space, $\card G = n$.

$\blacksquare$

Sufficient Condition

Let $\card G = n$.

From Sufficient Conditions for Basis of Finite Dimensional Vector Space, $G$ is a basis for $E$.

$\blacksquare$