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- #1

- Feb 5, 2012

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Here's a question and I'll also write down the answer for which I got zero marks. I would really appreciate if you can find where I went wrong.

**Question:**Let \(\phi,\,\psi\in V^{*}\) be two linear functions on a vector space \(V\) such that \(\phi(x)\,\psi(x)=0\) for all \(x\in V\). Prove that either \(\phi=0\mbox{ or }\psi=0\).

Note: \(V^{*}\) is the dual space of \(V\).

**My Answer:**Note that both \(\phi(x)\) and \(\psi(x)\) are elements of a field (the underlying field of the vector space \(F\)). Since every field is an integral domain it has no zero divisors. Hence, \(\phi(x)\,\psi(x)=0\Rightarrow \phi=0\mbox{ or }\psi=0\).