Let $V$ be finite dimensional Euclidean vector space. Let $a_1, \ldots a_m$ be nonzero vectors in $V$. Denote $H_i:=H_{a_i}$ be a hyperplane orthogonal to $a_i$. Set $X=H_1 \cap H_2 \cap \ldots \cap H_m$. Is it true that the orthogonal complement $X^{\perp}$ of $X$ is spanned by $\{a_1, \ldots a_m\}$?
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Finite-dimensional? – Titus Petronius Jun 11 '15 at 15:37
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Yes. I edited the original post. – user Jun 11 '15 at 15:44
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One has $X^\perp=H_1^\perp+\cdots +H_m^\perp$, but $H_i^\perp=\text{span}(a_i)$ by definition of $H_i$. So the answer to your question is yes.
