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 $E$ $\card G = n$.

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

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

Sufficient Condition
Let $\card G = n$.

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