# Cardinality of Generator of Vector Space

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## Theorem

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

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

Then:

- $G$ has at least $n$ elements.

## Proof

From Generator of Vector Space Contains Basis there exists a basis $B$ of $E$ such that $B \subseteq G$.

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

The result follows.

$\blacksquare$

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 27$: Theorem $27.14$ - 1996: John F. Humphreys:
*A Course in Group Theory*... (previous) ... (next): $\text{A}.2$: Linear algebra and determinants: Theorem $\text{A}.7$