# Double orthogonal complement of a finite module

Crossposted to mathoverflow

Let $W$ be the finite $\mathbb{Z}$-module obtained from $\mathbb{Z}_q^n$ with addition componentwise. Let $V$ be a submodule of $W$. Let $V^{\perp} = \{w \in W \, : \, \forall v \in V \quad v.w = 0 \}$ where "." is the dot product. Is it true that ${(V^{\perp})}^{\perp} = V$?

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I would use the notation $V^{\perp}$. –  Qiaochu Yuan Sep 12 '11 at 22:26
I don't think you waited long enough before crossposting. I would give it at least a day. –  Qiaochu Yuan Sep 13 '11 at 2:04