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Let an orthonormal basis of $\Bbb{R}^n$ be $\{e_1,\dots,e_n\}$, and $U$ be a subspace in $\Bbb{R}^n$. Can we construct the orthonormal basis of $U$ by taking randomly from $\{e_1,\dots,e_n\}$? If possible, would you prove it?

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  • $\begingroup$ Look up Gram-Schmidt algorithm. $\endgroup$ Jul 23 '20 at 15:31
  • $\begingroup$ @Chrystomath What if no $e_i$ belongs to $I$? $\endgroup$ Jul 23 '20 at 15:33
  • $\begingroup$ The subspace $U$ must be generated from some vectors $u_1,\ldots,u_k$. Apply GS to these. $\endgroup$ Jul 23 '20 at 15:35
  • $\begingroup$ Thanks. Is {u_1,...,u_k} a subset of {e_1,...,e_n}? $\endgroup$ Jul 23 '20 at 15:42
  • $\begingroup$ No, unless $U$ is generated by some axes of $\mathbb{R}^n$. $\endgroup$ Jul 25 '20 at 12:55
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In general, no, it is not possible. Take $n=2$, let $\{e_1,e_2\}$ be the standard basis of $\Bbb R^2$ and let $U=\{(x,x)\mid x\in\Bbb R\}$. Then no subset of $\{e_1,e_2\}$ is a basis of $U$, since neither $e_1\in U$ nor $e_2\in U$.

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