# Dimension of Algebraic Dual

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## Theorem

Let $G$ be an $n$-dimensional $R$-module.

Let $G^*$ be the algebraic dual of $G$.

Let $G^{**}$ be the algebraic dual of $G^*$.

Then $G^*$ and $G^{**}$ are also $n$-dimensional.

## Proof

Follows directly from Product of Linear Transformations.

$\blacksquare$

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 28$